University of Southern California Dissertations and Theses

Experimental study of radiometric forces with comparison to computational results

(USC Thesis Other)

Experimental study of radiometric forces with comparison to computational results

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content EXPERIMENTAL STUDY OF RADIOMETRIC FORCES WITH
COMPARISON TO COMPUTATIONAL RESULTS
by
Nathaniel P. Selden
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulllment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
May 2009
Copyright 2009 Nathaniel P. Selden
Acknowledgements
Without the unwavering love and support of my family, I am sure that I would have never
made it this far. For their devotion to the idea of letting me choose whatever path I may,
I owe a great debt to my parents. They are both amazing people, and I am very grateful
for the impact each has had on my personality, motivation, and ethics. To my siblings
(of which there seem to be an endless multitude): thank you all for your individual
viewpoints on life and happiness, they have greatly impacted my daily existence and I
am a better person because of each of you. Without hesitation, I include my ance Suzy
Perkins here, as her compassion and amazingly positive attitude have seen me through
many a long night in the lab. Thank you.
The guidance of many great scientists has also been greatly appreciated: Andrew
Ketsdever for his unfounded enthusiasm in molding me into a PhD candidate, as well
as his dedication to students in general... in this he is unmatched. Sergey and Natasha
Gimelshein for their daily interactions, their undaunted support, and numerous correc-
tions to my poorly formulated ideas. Dr. E.P. Muntz for his guidance, and his amazing
ability to always ask dicult questions, especially the ones I did not want to answer.
Many thanks are also due to Professors G. Shi
ett, H. Wang, J.A. Domaradzki, and M.
Gundersen for serving on my committee; each added something to the nished product,
ii
and have given me much to think about in terms of future scientic exploration. Finally,
I am very grateful to Dr. Ingrid Wysong at the Air Force Research Laboratory, without
whom none of this work would have been possible.
Thank you all, this research is dedicated to you.
iii
Table of Contents
Acknowledgements ii
List Of Tables v
List Of Figures vi
Abstract ix
Chapter 1: Introduction 1
1.1 Modern Importance of the Radiometer . . . . . . . . . . . . . . . . . . . . 1
1.2 History of the Radiometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Thermal Accommodation Coeecients . . . . . . . . . . . . . . . . . . . . 10
Chapter 2: Theory 13
Chapter 3: Experimental and Computational Setup 32
3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Computational Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Chapter 4: Gas Species: The Eects of Collisional Cross Section and Ac-
commodation Coecient 56
Chapter 5: Geometry Considerations: Area, Perimeter, and Attachment
Eects 63
Chapter 6: Volume Considerations: Chamber Eects 77
Chapter 7: Multiple Vanes: A Preliminary Study of Force Optimization 83
Chapter 8: Conclusion 93
References 97
iv
List Of Tables
1.1 Various historical values of accommodation coecient . . . . . . . . . . . 11
4.1 Comparison of experiment and computation for Ar, He, and Xe . . . . . . 61
4.2 Comparison of computed values of accommodation coecient with various
sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1 Comparison of solid and hollow radiometer vanes . . . . . . . . . . . . . . 65
v
List Of Figures
1.1 Modern example of a Crookes Radiometer (Courtesy [18]) . . . . . . . . . 1
2.1 Simple model of a radiometer in a chamber . . . . . . . . . . . . . . . . . 14
2.2 Change in force over full range of alpha for various surface temperatures . 18
2.3 Comparison of theoretical approximation of radiometer force. . . . . . . . 24
3.1 CAD drawing of a radiometric device . . . . . . . . . . . . . . . . . . . . . 33
3.2 Comparison of three experimental radiometer vanes . . . . . . . . . . . . 34
3.3 Radial temperature prole along both faces of a radiometer . . . . . . . . 36
3.4 Schematic of a thrust stand . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Cutaway view of a LVDT (courtesy of [61]) . . . . . . . . . . . . . . . . . 38
3.6 Cross sectional schematic of electrostatic calibration combs . . . . . . . . 40
3.7 Calibration results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.8 Initial conguration of 0.4m chamber . . . . . . . . . . . . . . . . . . . . . 44
3.9 Representative trace of ltered data, background pressure 0.52 Pa. . . . . 45
3.10 Initial conguration of 3m chamber. . . . . . . . . . . . . . . . . . . . . . 47
3.11 Impact of gas
ow on thrust measurement. . . . . . . . . . . . . . . . . . 48
3.12 Conguration of ISO 400 tube in 3m chamber. . . . . . . . . . . . . . . . 50
3.13 Attachment mechanisms designed to minimize impact . . . . . . . . . . . 51
vi
3.14 Comparison of nal four experimental radiometer vanes . . . . . . . . . . 52
3.15 Changing force as a function of temperature dierence across surfaces . . 53
4.1 Computed force for various gases . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Normalized force vs Knudsen of various gases . . . . . . . . . . . . . . . . 58
4.3 Experimental results for various gases in original small chamber . . . . . . 59
4.4 Experimental results for various gases in large chamber . . . . . . . . . . 60
4.5 Comparison of experimental and axissymmetric computational helium re-
sults for 3.0m chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1 Pressure on the main surfaces of the radiometer . . . . . . . . . . . . . . . 64
5.2 DSMC (top) and ES-BGK (bottom) streamlines and temperature eld in
helium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 Computed pressure dierence along the plate . . . . . . . . . . . . . . . . 67
5.4 Normalized force for dierent device geometries in the 0.4m chamber . . . 68
5.5 Numerical results for two dierent plates . . . . . . . . . . . . . . . . . . . 69
5.6 Comparison of experimental and computational results . . . . . . . . . . . 70
5.7 Comparison of experimental results for 3 attachment mechanisms. . . . . 71
5.8 Measured force on three dierent plates in argon . . . . . . . . . . . . . . 74
5.9 Comparison of force ratios for 2 geometry pairs . . . . . . . . . . . . . . . 76
6.1 Temperature gradient in the gas as a function of chamber . . . . . . . . . 78
6.2 Force as a function of chamber size . . . . . . . . . . . . . . . . . . . . . . 79
6.3 Comparison of device geometries in the 3.0m chamber (CHAFF) . . . . . 80
6.4 Comparison of device geometries in the 3.0m chamber (CHAFF) . . . . . 81
6.5 Ratio of curve t data for a given device geometry . . . . . . . . . . . . . 82
vii
7.1 ES-BGK streamlines and density eld in argon (0.5Pa) for multiple con-
gurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.2 ES-BGK temperature eld in argon (0.5Pa) for multiple congurations. . 87
7.3 ES-BGK normalized pressure eld in argon (0.5Pa) for multiple congu-
rations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.4 ES-BGK normalized dierential pressure prole in argon (0.5Pa) for mul-
tiple congurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.5 ES-BGK vane force for multiple pressures and congurations. . . . . . . . 90
7.6 ES-BGK force per unit length of vane for multiple pressures and congu-
rations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
viii
Abstract
A study of the radiometric forces on heated plates has been conducted both experimen-
tally and computationally. The experiments were carried out at USC in two vacuum
chambers up to a maximum pressure of 6 Pa for various carrier gases. The computations
were performed with both the DSMC and ES-BGK methods for a 2-D gas
ow over a
comparable range of pressures. It is shown that the radiometric devices provide maxi-
mum force at a Knudsen number approximating 0.1. Of the various gases tested, helium
provides the largest peak force. Qualitatively, the experimental data and computational
results are similar. A lack of experimental data on gas-surface accommodation and
ow
three-dimensionality yields up to a 40% dierence in the magnitude of the measured and
computed forces, but it is shown that this discrepancy can be used to predict accommoda-
tion values. Comparison of four geometric congurations has shown that the eect of the
area is signicant at pressures up to where the force is maximum. It is also demonstrated
that the size of the chamber in which the radiometer resides is of primary importance,
where the chamber dimensions are inversely related to the generated force. Finally, sim-
ulation of multi-vane congurations have shown that the optimal spacing of vanes can be
tailored for specic uses; for maximum force production a tight spacing should be used,
while maximum eciency requires spacing on the order of a vane dimension. While the
ix
results so far are encouraging, they are far from complete. Further improvements would
include: a new experimental setup to reduce uncertainty with highly accurate tempera-
ture control and measurement, an in situ way to prepare the surface as well as measure
its cleanliness, and an in depth iterative computational study observing the impact of
multiple radiometer vanes at numerous seperations.
x
Chapter 1
Introduction
1.1 Modern Importance of the Radiometer
Figure 1.1: Modern example of a Crookes Radiometer (Courtesy [18])
For more than 125 years the Crookes Radiometer (Fig. 1.1) has fascinated everyone
from scientists to school children. In the second half of the last century scientic experi-
mentation began to wane, likely due to a lack of an obvious application. The minuscule
1
interest in the problem was reserved for a few research papers published on the sub-
ject, with other publications being historic analyses and overviews. The situation began
to change in the last two decades, where the advent of microelectromechanical systems
(MEMS) fabrication and the quest for a viable near-space propulsion technology have cre-
ated renewed interest in radiometric forces. The possibility of using modern technology
to enhance, or more eciently utilize, these forces has led to proposed novel applications
at both the micro- and macro-scales.
In the micro-scale realm, the extremely small distances involved allow these devices
to operate in the transitional or free-molecular regimes at near-atmospheric pressures. It
is under these conditions that thermal gradients across the sides of dierentially heated
surfaces, and also in the gas, can create observable radiometric forces. Wadsworth[56] has
shown them to be applicable to the modern microactuator, where the direct simulation
Monte Carlo (DSMC) method was used to model forces on vanes mounted on an armature.
This method, along with experimental measurements, has been employed by Ota[35] et
al to study a concept of an opto-microengine which utilizes a laser to create the required
thermal gradients. Subsequently, a series of papers by Passian [38], [36]with co-workers
have been published , where radiometric phenomena were studied experimentally and
analytically, mostly with application to microcantilevers. The prime importance of this
work was to address the concern that these forces may have a detrimental eect on modern
microscopy by adding unwanted de
ection to the measurement device as it warms up in
a reduced pressure environment.
With regard to the macro-scale, the larger dimensions require that low pressures be
attained to reach the regimes where radiometric forces begin to be observable. This can
2
be achieved in a laboratory vacuum system, where it has been suggested by Passian[37] et
al in a third paper that the use of radiometric forces may provide an approach to study
gas-surface translational energy accommodation. Outside of the laboratory, the upper
atmosphere provides a possible environment for a new concept of a high-altitude aircraft.
Such a craft has been put forward by G. Benford[4], which would be supported by ground
based microwave energy utilized to create radiometric eects. Finally, and perhaps of
most interest, is an unpublished presentation to the Department of Energy[42] suggesting
that a macro-scale device with micro-scale features could be used at atmospheric pressure
to create large usable forces. In this presentation it was suggested that such a device would
consist of numerous holes with diameters on the order of the molecular mean free path
(), eectively maximizing the perimeter over which the radiometric force could act.
The new studies of the old radiometric phenomena have been supported on one hand
by modern technologies that allow more accurate measurements, and on the other hand
by state-of-the-art numerical methods that rely heavily on parallel computing. These
two factors, along with the revived interest in the application of radiometric phenomena,
have prompted a revisiting of the radiometer. The research reported in this thesis was
focused on several topics of interest; the rst was a study of the contribution of the
"collisionless" (area) versus the "collisional'" (edge) forces to the total radiometric force.
The objective of this study was to examine experimentally radiometric forces created
by rareed gas
ows on heated plates of dierent shapes, and to analyze numerically the
change in the total force as a function of gas pressure and chamber volume. Experimental
measurements of radiometric forces have the benet of correctly accounting for dierent
factors, such as gas-surface accommodation, internal structure of molecules, and complex
3
three-dimensional geometries. Numerical modeling provides detailed information of gas

ow properties, including the in
uence of important surface phenomena.
Another important subject of interest is the chamber size eect. It is clear that the
size of the chamber will have an important impact on both the
oweld and the tempera-
ture gradient created by the radiometer. Historically this eect has largely been ignored,
where it was commonly assumed that the chamber was much larger in dimension than a
vane, and many free paths distant. Here, an attempt has been made to study not only
the limiting cases of both extremely large and extremely small chambers, but also several
intermediate cases. As comparison of the numerical simulations and the experimental
data is of prime importance, a third topic of research was to obtain accurate accommo-
dation coecients for the experimental device. Finally, in an attempt to understand the
feasibility of using the radiometer for practical application at high pressures, a study was
undertaken to determine how vane separation aects force production. Ultimately, the
goal of this thesis is to further explore the physics of the radiometer with the hope that a
new understanding of the venerable device can aid in bringing it from an obscure novelty
to a practical energy conversion device.
1.2 History of the Radiometer
The repulsion and attraction of bodies induced by radiation received a great deal of
attention from a number of prominent scientists in the 19th and 20th centuries[59]. The
rst published experiment was conducted by Abraham Bennet[5], who reported in 1792
the negative result of light shone on a paper vane, which was suspended by a ber thread
4
in vacuum. At that time, he was unable to see any motion distinguishable from the eect
of heat. The rst successful experiment was conducted by Fresnel[15] who observed in
1825 a repulsion between two suspended foil vanes when sunlight was focused on them in
a low-pressure container. In the 1870s, William Crookes proposed several dierent types
of apparatus to investigate the radiometer eect[10, 11]; one of them became known as the
Crookes radiometer. It consists of an airtight glass bulb containing a partial vacuum with
a set of half-blackened vanes mounted inside the bulb on a spindle; the vanes rotate when
the dark side is exposed to light or another heat source. Crookes incorrectly suggested
that the force causing the vanes to move was due to photon pressure. This theory was
originally supported by J.C. Maxwell[31] who had predicted this force, but was quickly
proved incorrect. While his initial hypothesis was incorrect, Crookes' device helped lay
the foundation of more than a century of progress in gas kinetics, and was an important
means to test a variety of new theories[57] about the sources of these phenomena.
While the bulk of the research in the decades immediately following this initial work
focused on the interaction of the gas molecules with the radiometer vane, comparatively
little work was done on the interaction of the entire system with the walls of the vacuum
vessel. O. Reynolds initially proposed a reasoning based on surface outgassing, and then
presented a more rigorous explanation based on kinetic theory[39]. According to the
latter theory, the gas in the partially evacuated bulb is the main driving force responsible
for the rotation of the vanes. Reynolds also took part in the experiments conducted by
Schuster[44] that provided the rst experimental evidence that gas-related forces were
the dominant cause of the radiometric eect. In this experiment, the radiometer case was
suspended by parallel bers and light was directed onto the vanes. The radiometer case
5
was pushed in the direction opposite the vanes, proving that the radiometric phenomenon
is caused by the interaction between the heated side of the vane and the gas. Despite this
discovery that the container was also aected by the moving gas, another 40 years passed
before any focused experimental work was done with specic regard to the chamber's
contribution to the radiometer.
The kinetic theory explanation given by Reynolds is in fact a free molecule(or collision-
less) approximation of the radiometric eect rst deduced by Stoney[50]: the molecules
leaving the hot side depart with an increased velocity relative to those leaving the cold
side. Here, collisionless simply implies a condition in which no intermolecular collisions
occur between gas molecules, but molecules are free to collide with the vane surfaces.
This leads to a larger momentum exchange with the hot side, and results in the motion
of the vanes with the hot side trailing. The situation is, however, dierent in transitional
or near-continuum
ow. According to Reynolds[40], the molecules with higher velocities
leave the hot side of the vane and collide with incoming molecules. These collisions cut
the surface
ux more eciently than those re
ected on the cold surface. Essentially, this
means that these eects compensate each other, and pressures in the center of the vane
are equal. At about the same time, Maxwell[32] also showed that an unbalanced force
exists near the edge of the heated side of the vane, where the temperature in the gas is
non-uniform.
As with many important advances in science, West[57] stumbled onto his contribution
in 1919 while exploring an entirely dierent topic. Seeking (like Crookes before him) to
measure the pressure of light, he found (as had many others) that his measurements were
disturbed due to the presence of a temperature gradient in his experiment. Noting that
6
the disturbances grew in size as his piece of foil was moved further from the centerline of an
evacuated tube, he proceeded with a series of experiments designed to explore interaction
with the container. He surmised that while radiometric forces should be minuscule in his
apparatus due to the temperature uniformity of his foil, the asymmetry of the experiment
itself created non-equilibrium conditions in the gas. Ultimately he concluded with three
main ideas: at the lowest background pressures the force on the foil varied with the square
root of gas temperature, that thermal transpiration was the cause of the disturbances at
higher pressures, and nally that the gas gradient along the face was signicantly more
important than the surface gradient across the edge. As a point of reference, a plot of
his demonstrates that at higher pressures halving the distance to the vessel wall doubled
the force on the foil strip.
An even more exhaustive study was undertaken by Brueche and Littwin [9] in the late
1920's. In this comprehensive work an extremely large experimental matrix was covered;
this included a variety of gases, radiometer geometries, and separation distances. One of
the main conclusions of the paper was a formula for the relationship between force and
pressure showing an almost Gaussian response to pressure, modeled empirically as
F =
1
(a=p) + (p=b)
;
where a and b are curve tting parameters specic to an experimental data set. This
expression re
ects the fact that the radiometric force has a maximum at some pressure
that depends on gas and geometric properties, which was shown quantitatively as early
as 1919[58]. A second conclusion, and one of great relevance to the understanding of the
7
impact of the chamber wall, was verication of West's nding that at high pressures the
force on the vane scales with the inverse of the distance.
It is useful to explore why this is so. Under collisionless conditions, the radiome-
ter has an extremely large force production potential. It has been shown by multiple
experiments[58],[9] that an increase in pressure will result in a proportional increase in the
force. If maintaining these conditions were possible and the linear increase were allowed
to continue, the output of a radiometer would reach forces of approximately 75N/m
2
K
at an atmosphere of background pressure. In reality, the onset and eventual dominance
of molecular collisions at higher pressures causes the force production to rapidly decline.
This force decay can be slowed by moving the vane location closer to the chamber wall.
In this manner, every halving of the distance to the wall also halves the number of colli-
sions of a molecule traversing the space between wall and vessel. Taken to the limit of a
mean-free path of separation, a majority of the molecules impinging on the vane would
in eect behave as if they were collisionless with respect to other molecules.
It was several years later (almost fty years after Maxwell suggested the existence of
an edge force) that Einstein presented a simple theory that related the force on the vanes
to their perimeter. This edge dependence of the vane force found partial conrmation
in experimental work[30], where the force was found to depend on perimeter length, al-
though not to the extent that Einstein had predicted. Where the inversely proportional
pressure dependence of the radiometric force of a vane placed in a temperature gradi-
ent, derived by Einstein, is similar in magnitude to the high-pressure part of the general
dependence proposed by Bruche and Littwin, Loeb[28] suggested that it failed qualita-
tively in describing the precise dependence on pressure that is experimentally observed.
8
Since about that time, the edge theory has become widely accepted. At low pressures,
a free-molecular area force is the dominant one, with the force increasing with pressure.
At high pressures, the collisional edge force becomes dominant, and the force decreases
as pressure increases.
The strong interest in the radiometer problem since 1873 declined steadily after 1928,
mostly because the issue of force production was considered closed, and no direct appli-
cation for radiometric forces had been identied. There was, however, additional research
added to the body of work describing the motion of the radiometer. In a section in his
book published in 1938, Kennard[25] suggests that thermal creep along the side of the
radiometer creates a tangential force along the edge surface. He also derived a formula
suggesting that this force was linearly dependent on the temperature gradient along the
wall, but inversely proportional to the pressure. His theory found relative conrmation
by Sone[47] several decades later, who also showed that the velocity of the molecules
moving along the wall was also dependent on the gradient of the wall.
The current understanding of the phenomenon that drives Crookes' radiometer has
been summarized by Draper[12]: a temperature gradient exists on the surface if tangential
stresses are to arise. These stresses are the result of thermal transpiration, with the gas
moving over the surface from the cold to the hot side. Following this explanation, the
principal force that contributes to the rotation of the vanes in the pressure regime where
the radiometer is most eective, is the force created near the edges (a zone with the
dimensions of a mean free path, according to Einstein).
9
1.3 Thermal Accommodation Coeecients
In the late 1950's and early 1960's a concerted eort took place to identify a more physi-
cally realistic model than the classical one proposed by Maxwell[32]. With the arrival of
the molecular beam, numerous researchers began experimenting with molecules having
precisely controlled velocity distribution functions. Utilizing the fact that the incoming
distribution function was described by the design of the molecular beam, it seemed that it
would be fairly straightforward to determine the only remaining unknown: the re
ected
distribution function. Useful as the experiments were, they ultimately failed at producing
a cohesive theory explaining the precise interaction of a gas and a surface.
An attempt at developing a model of surface interaction was proposed by Stickney
[49] in 1962, and is simply a modied version of Maxwell's model. Based on an earlier
paper [22] showing diuse re
ections from metallic surfaces, he suggests that some frac-
tion " are completely accommodated while the remaining fraction 1-" are re-emitted
with random direction and no energy accommodation. While similar in notation, it is a
strikingly dierent model because it removes the possibility of specular collisions. In this
same paper he reports experimental data for measured relative force on a heated plate
as a function of surface temperature. Using a well dened normalization procedure, he
shows that while increasing the surface temperature increases the measured force on the
heated plate, it does not appear to change the normal momentum accommodation of the
gas to the surface. He concludes with the idea that surface contamination is more re-
sponsible for accommodation than the substrate itself (so long as it is a relatively smooth
10
substrate). Several gases and surfaces were tested, and the results of these tests appear
in table 1.1.
Surface Gas Temp (K) α Experimentalist Year
Aluminum Argon 300 0.826 Teagan 1966
Helium 298 0.58 Alofs 1971
Helium 550 0.6* Stickney 1962
Hydrogen 600 .75* Stickney 1962
Nitrogen 298 0.82 Alofs 1971
Nitrogen 300 0.76 Teagan 1967
Nitrogen 550 .9 - .95* Stickney 1962
Stainless Helium 298 .4 - .58 Alofs 1971
Nitrogen 298 .7 - .82 Alofs 1971

Table 1.1: Various historical values of accommodation coecient
In the early 1970's scientists began looking at the local density of a gas near a surface.
In a paper published about that time [2], electron beam induced luminescence was used to
precisely measure the density of both helium and nitrogen between two parallel surfaces
of dierent temperature. By varying the distance of the electron beam from each of
the surfaces, they were able to create a relatively complete density prole of the gas.
Using the theoretical models of Liu and Lees, the authors found satisfactory agreement
between their experimental values and those of the model, with the exception of the data
nearest the hot plate. Using these density proles and a method presented by Teagan
and Springer [52], [51], they also presented values for the accommodation of these gases
onto both aluminum and stainless steel surfaces (see table 1.1).
In a comprehensive summary of the work done by Lloyd B. Thomas between 1937
and 1989, H.Y. Wachman demonstrates how Thomas proved beyond a doubt that surface
contamination played a very important role in the measurement of the accommodation
coecient.[54] The summary also shows that so long as the surface is maintained without
11
any contamination, there is no correlation between pressure and accommodation. It con-
tinues with a comparison of a variety of experiments in which surface conditions were not
maintained, where it can be seen that uncontrolled surface conditions yield an extremely
wide range of accommodation coecients for a variety of surface/gas combinations. The
summary nally concludes with plots demonstrating the time dependent nature of adsorp-
tion, the increase in accommodation with surface coverage, and the eventual constancy
of the coecient once a monolayer of gas has been formed.
While a deep knowledge of the accommodation coecient is vitally important to a
thorough understanding of the radiometer, there is an extremely large body of work
suggesting that it is dicult to obtain accurately[41]. Compounding this diculty is
a similarly wide variety of approaches to incorporating it into a practical wall collision
model. As the greater goal of the present work is to explore the fundamental gas
ow
phenomena surrounding a radiometric device, it is important to characterize the eect
that the accommodation coecient has on device operation. As the devices used do
not lend themselves readily to surface preparation, the experimental work was directed
instead at maintaining a high level of consistency. This ensured that comparison between
devices and data repeatability led to valuable insights about device performance, and to
a direct comparison of specic surface conditions to computational results.
12
Chapter 2
Theory
The simplest way to mathematically describe the force produced by a radiometer is to
begin in the free-molecule(FM) regime. In this regime the incoming velocity distribu-
tion to the radiometer surface is wholly unaected by the velocity distribution of any
molecules re
ected from the surface, as collisions between molecules are assumed never
to occur. In reality, such a condition never quite occurs, but when the gas is suciently
rareed the approximation is quite accurate. The FM regime is generally used as the low-
pressure limiting case for any gas kinetic mathematical model. For a reasonable amount
of gas rarefaction to be present, the Knudsen number (Kn) of the system must be large.
Generally, it must hold that Kn10, where the Knudsen number is dened as
Kn =

L
(2.1)
Here is the mean free path of a gas molecule and L is a characteristic dimension of the
system.
13
In a system where these conditions hold, the radiometer can be modeled as an innitely
thin plate with dimension L and area A in a chamber with dimensions much larger than L.
This setup is shown in Figure 2.1, whereP
0
andT
0
represent the pressure and temperature
of the freestream gas, and T
HOT
and T
COLD
represent the hot and cold temperatures of
the radiometer surfaces.
Figure 2.1: Simple model of a radiometer in a chamber
As it was assumed that the
ow is collisionless, the incoming velocity distribution can
be represented as f
i
(T
0
), while the re
ected distributions are represented as f
r
(T
HOT
)
and f
r
(T
COLD
). Under these conditions it is straightforward to show for an equilibrium
distribution function that the momentum
ux of the incoming molecules to both the hot
and cold surface are identical, and thus completely cancel each other out. Looking at the
re
ected momentum
uxes for a stationary plate perpendicular to the u direction, the
equation is found to be
14
p
r
=n
r

3
2
Z
1
0
Z
1
1
Z
1
1
(mu)ue
(u
2
+v
2
+w
2
)
dudvdw (2.2)
Here n
r
is the number density of the gas very near the surface, m is the mass of the gas
molecule, and is dened as
=
m
2kT
: (2.3)
This equation simplies to
p
r
=
n
r
kT
r
2
(2.4)
To ensure conservation of mass, the incoming mass
ux has to be equal to the mass
ux
re
ected from the surface:
_
N
i
=
_
N
r
(2.5)
The
ux to a stationary surface in a gas with zero
ow velocity can be represented
by the equation
_
N =
n

c
0
4
(2.6)
where

c
0
is the mean thermal speed represented by

c
0
=
r
8KT
m
(2.7)
For the re
ected gas modeled as a completely accommodated wall gas, it can be shown
that
15
n
r
=n
0
r
T
0
T
r
(2.8)
Substituting into equation 2.4, it can be seen that the re
ected pressure is solely a
function of the incoming number density, incoming temperature, and the temperature
of the surface. As the force produced by the radiometer is simply the dierence in
pressure between the two sides of the device multiplied by the area, the equation for force
production in the free molecular regime is
F =
P
0
2
A
"
r
T
H
T
0

r
T
C
T
0
#
(2.9)
In reality it is highly unlikely that complete accommodation with the surfaces will
occur. The most widely used historical model for incorporating surface energy accom-
modation is that of Knudsen[26], where (again assuming no intermolecular collisions and
a diusely re
ected equilibrium velocity distribution) the accommodation coecient is
given by
=
T
0
T
r
T
0
T
s
(2.10)
To clarify this equation, it can be seen that the top half describes how much thermal
energy the gas acquired, while the bottom half describes the amount of energy available
if the gas completely accommodated to the temperature of the surface. It is also worth
noting that this equation is only valid in cases with a very large container, where it is
assumed that many collisions with the container walls occur before a molecule returns to
16
the heated surface. Rearranging the equation, it can be seen that the temperature of the
re
ected gas is entirely a function of the initial conditions and the extent to which the
incoming molecules interact with the surface.
T
r
=(T
s
T
0
) +T
0
(2.11)
Substituting re
ected temperatures into eq. 2.9 in place of the surface temperatures,
the force produced by a stationary radiometer vane in a non-
owing gas for the generalized
case of non-unity accommodation is
F =
P
0
2
A
2
4
s
(T
H
T
0
) +T
0
T
0

s
(T
C
T
0
) +T
0
T
0
3
5
(2.12)
where it is assumed that both the hot and cold surfaces have an identical .
A plot of this result is shown in Fig. 2.2 for a variety of temperatures, where the
value shown in the plot is the surface temperature of the hot side. For the purpose of
comparison, the theoretical temperature dierence between the hot and cold sides was
40K, and the ambient temperature was set to be 300K. This plot shows that the force
produced by the radiometer should vary with accommodation in a nearly linear fashion
so long as the temperature dierence between the ambient gas and vane surfaces remains
relatively small. At very large dierentials(
T
HOT
T
0
> 100), the force appears to vary with
p
.
Although the assumptions made in the previous derivations are a convenient aid to
understanding the radiometer, they are inadequate for any device operating at pressures
above the largest degrees of rarefaction. As the force production of the radiometer peaks
17
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Alpha
Normalized Force
400K
800K
1600K
3200K
Root(alpha)

Figure 2.2: Change in force over full range of alpha for various surface temperatures
in the transition between a Knudsen gas and the continuum regime, these formulae quickly
become limiting cases at the low end of the pressure spectrum. To complicate matters
further, several other notable
ows appear in this regime: thermal creep[25, 47], thermal
stress[27], and second-order slip[47]. All things considered, the search for an analytical
solution becomes dicult.
When searching for a more practical approach, two tools present themselves: ap-
proximation and computation. While each of these methods has advantages, each are
also plagued with drawbacks. It is illustrative to begin with approximations derived by
scientists during the last century, and then to move into the relatively more recent compu-
tational theory. The earliest of these approximations date back to the 1920's when both
Einstein[13] and Epstein[14] published papers on the topic. They each had fundamen-
tally dierent approaches; Einstein proceeded to explore the higher-pressure operation
of the radiometer with his traditional thought experiments, while Epstein used a more
mathematically rigorous approach which relied on thermal creep.
18
In Einstein's approach (as summarized by Loeb in 1927[28]), he began with the knowl-
edge that in a continuum gas, pressure is equal everywhere. From this, he surmised that
there were basically three regions of importance: 1) the hot gas directly above the surface,
2) the cold gas far from the edge, and 3) the region in which these two bounding regions
must interact. If this interaction region is assumed to be in width at the edge of the
radiometer, the uniformity of pressure does not hold as the cold region transitions to the
hot one. Under these conditions, a pressure normal to the vane will exist along the edge
of the device, where Einstein calculates its value per unit length as
K =
f
u
=
1
2
p

2
T
@T
@x
(2.13)
where
@T
@x
is the temperature gradient along the edge of the radiometer, p is the back-
ground pressure, is the mean free path, and T is the temperature of the gas at the last
collision before it impacts the surface.
Though he did not give the derivation, and suggests that it is only applicable as
an order of magnitude approximation, Einstein's nal equation for a radiometer vane
warmed on one side (per unit length of edge) is
K =
f
u
=p
T
T
(2.14)
Here, T is the temperature dierence between the unheated surface and the heated
one. Loeb goes on to suggest that under these conditions force production should be
independent of pressure, as pressure has an inverse relationship with the mean free path.
A deeper reading of Einstein may suggest otherwise however, as he denes T as the
19
average temperature the gas obtained from its most recent collisions. Though a subtle
change, this has enormous eect on the predictions made by the equation. Simply put,
the temperature term in the equation becomes pressure dependent. Fundamentally this
makes sense, because as pressure increases the most recent collision will be ever closer to
the heated surface, and thus at a higher temperature.
While Einstein's approach may yield appropriate approximations at high enough pres-
sures where the last collision occurs at a temperature very near the surface temperature,
it does not work particularly well in the transition regime where the gas temperature
obtained by the last collision is somewhat lower than that of the vane. This is not neces-
sarily due to a failing of the theory, but more so to the fact that it is dicult to accurately
approximate the temperature of the gas.
In what seems like a contrasting view of the mechanism behind the operation of the
radiometer, Kennard[25] suggests (in summary of Epstein) that
"`Qualitatively the creep theory of radiometric action is completely suc-
cessful. A quantitative calculation of the force, however, presents, unfortu-
nately, a dicult problem...The complete problem has been solved only for
the ideal case of an ellipsoidal disk, circular in principal outline but of ellipti-
cal cross section, which, if thin, should present some approximation to a
at
disk."'
He shows that the equation that Epstein ultimately arrived at is
F =3
R
2
p
T (2.15)
20
where R is the gas constant of the gas, is the gas viscosity, and T is one of the
following functions. If a temperature gradient exists, where the uniform gradient
dT
dx
at
large distances is parallel to the axis of the disk, then the value of T is
T =
4

a
1 +
2

k
d
kg
a

dT
dx
: (2.16)
Alternately, if a uniform beam of light impinges on the surface with an intensity J,
he found that
T =
2

aJ
k
g
+
2

k
d
a

; (2.17)
where J is measured in ergs/cm
2
/sec, a is the radius of the disk and its thickness, andk
d
andk
g
are the conductivities of the disk and of the surrounding gas, respectively. Epstein
further simplies the expressions under two limiting conditions; the rst of these being
a nonconducting disk, while the second is a highly conducting(extremely thin) one. For
a nonconducting vane illuminated by a uniform beam of light, eq. 2.15 can be combined
with eq. 2.17 to yield
F =
6R
2
aJ
pk
g
: (2.18)
For a highly conductive vane, eq. 2.15 can instead be combined with eq. 2.18 to yield
F =3
R
2
J
pk
d
: (2.19)
21
It is easily observed that much of this theory was worked out to facilitate theoretical
prediction of radiometric forces without temperature measurement. If the actual temper-
ature dierence between the hot and cold sides of the plate is known, T in Equation
2.15 can simply be replaced with this value.
In a more recent paper, Scandurra [43] uses the Chapman-Enskog method to calculate
both normal and shear stresses imparted on the radiometer. In a manner similar to that
of Einstein, the paper treats the vane as a series of three regions in which the pressure
is equal everywhere. The regions are essentially the same as Einstein's, with the region
above the vane being uniformly hot, the region far from the edge being cool, and the
region in-between being the place where the transition occurs. Unlike Einstein he treats
the gas kinetics using a much more mathematically rigorous analysis, and instead of using
a very simple model for the distribution function, he uses
f(~ v) =

3=2
e
(u
2
+v
2
+w
2
)

1 +Av

5
2
(u
2
+v
2
+w
2
)

; (2.20)
where
=
m
2kT
; (2.21)
and
A =
15
32n
2
T
r
m
kT
dT
dy
: (2.22)
In the above equations, u, v, and w are the velocity components in the x, y, and z
directions, m is the mass of a gas particle, k is the Boltzmann constant, and
2
is the
22
molecular cross section of a hypothetical hard sphere molecule. Using his analysis, and
assuming the force is produced in an area one mean free path wide, he nds that
F
normal
=
15k
32
p
2
2
T

l (2.23)
in the direction opposite the gradient, where is the thickness and l is the perimeter of
the vane. Using similar logic, he also lists the shear force in the same direction as
F
shear
=
15k
64
p
2
2
T

(l); (2.24)
or
F
shear
=
15k
64
p
2
2
T

(l); (2.25)
when . In general he places several limitations on the use of these equations, the
simplest of which state that the size of the device has to be much greater than a mean
free path, the thermal gradient must be small to moderate, and the molecules must be
assumed to be fully accommodated to the surface. Mathematically he denes the upper
limit of the Knudsen number as
15
16
Kn
T
T
(2.26)
and the thermal gradient as
15
16

dT
dy

T
T
(2.27)
Beyond the two most widely accepted approximations of Einstein and Epstein, as well
as the extension of Einstein by Scandurra, there are quite a few other approximations that
23
were published by various scientists. Some are only dierent from those presented in their
calculations of coecients, while others are dramatically dierent in both mathematical
approach as well as functional dependence. As a complete summary of this long list of
published work has been done elsewhere[25, 29], it seems more appropriate to summarize
the large dierences in approximation graphically, while also noting the nal equation
and the person who derived it. A plot of each prediction is shown in Fig. 2.3, where it
is clear that the majority of these approximations yield results that are inconsistent with
each other. To obtain these curves a single set of constants was chosen which re
ect the
conditions present in the experimental work, where the value of dT/dx used is similar to
that of the experimentally measured temperature dierence between the vanes, and the
vane and chamber dimensions were used where appropriate.
1.00E-09
1.00E-08
1.00E-07
1.00E-06
1.00E-05
1.00E-04
0 1 2 3 4 5
Pressure, Pa
Force / ΔT, N/K
FM Theory
Scandurra
Einstein
Sexl
Epstein
Hettner

Figure 2.3: Comparison of theoretical approximation of radiometer force.
Here, the largest force is predicted by free molecule theory (Eq. 2.9), where collisions
between molecules are completely ignored. The second highest force plotted are the
24
results of Scandurra, which combine both the shear and normal forces of his predictions.
In addition to the results of Einstein and Epstein, Sexl[46] presents the equation
F
Disk
=
14:72
n + 5
p
2
T
T: (2.28)
which is almost identical in nature to that of Epstein, though with a dierent coecient.
Finally, results are plotted for Hettner and Czerny's[20] estimate of the shear stress
produced at the edge of the vane
P
12
=
3
4

2
aT
T
x
: (2.29)
where a is the distance from the vane edge to the chamber, and and are viscosity and
density, respectively.
The important thing to note about the above plot is simply the fact that the results
are not particularly self consistent. Though they do exhibit similarity in functional shape
(with the exception of Einstein), there are several orders of magnitude between their
predicted forces. It is precisely this dierence that makes other methods of approximation
both interesting alternatives, and necessities for further insight.
As it has been demonstrated that mathematical approximation up to this point has
not yielded extremely satisfying results, it is worth introducing a second tool for predicting
radiometer forces. This method is computational simulation, where the modern ability
of computers to perform immense numbers of calculations in rapid succession has led to
25
interesting and previously unattainable results. The backbone of such a simulation in gas
kinetics is the Boltzmann equation[19], where
@f
@t
+ v
1

df
@q
1
=
Z
d!dv
2
V [f(q
1
; v
1
;t)f(q
1
; v
2
;t)f(q
1
; v
1
;t)f(q
1
; v
2
;t)]: (2.30)
Here,
@f
@t
is the change in the distribution function with time while v
1

df
@q
1
represents
a convection term. It should be noted that a third term would be present on the left
hand side if external forces such as gravity were present, but for the case at hand they
are presumed to be non-existent. The integral on the right hand side of the equation
is called the collision integral, and the distribution function is shown to be a function
of molecular position (q), velocity (v), and time. The bar over the velocities indicate
that these are post-collisional values, while the vectors without the bars are the values
of the velocity before a collision has occurred. The d! term is a function of the impact
parameter, which in general is dependent on the nature of the intermolecular potential
and the relative speed of the colliding pair. When the collision integral is taken as a whole
it represents the sum total of all collisions occurring in velocity space. The equation is
exact under certain limitations, where its applicability is valid so long as:
1. The assumption of molecular chaos holds
2. Three-body collisions can be neglected and binary collisions dominate
3. Gradients in the gas are large enough that collisions can be thought of as localized
in physical space.
26
Solving this equation in its exact form can only be done for the equilibrium case when the
distribution function is not changing in velocity space (
@f
@t
= 0). This solution leads to
what is known as the Maxwell distribution, which is usually labeled as f
0
and represents
the exponential function
f
0
(u;v;w) =

3=2
e
(u
2
+v
2
+w
2
)
; (2.31)
where
=
m
2kT
: (2.32)
There are a multitude of extensions that have been worked out for the Boltzmann
equation, but they invariably involve limited cases where the boundary conditions are
precisely prescribed and an approximate molecular interaction model is used. Despite
these shortcomings, an enormous amount of literature has been dedicated to the proof
and application of several successful methods. The rst of these to be discussed is direct
simulation Monte Carlo, otherwise known by its acronym DSMC. The primary use of
DSMC is credited to G. Bird[7], where he laid out the foundation of the approach in
1976. As he dedicated a chapter of his book to the method, and many scientists since
have written volumes about the work, the reader is referred to the literature for an in-
depth analysis. A summary of the method is however, useful in understanding why it is
used and how it diers from alternate methods.
Simply stated, DSMC is a particle method which relies heavily on statistical simulation
of molecular collisions. To achieve this, a computational domain is created by structuring
a grid in coordinate space in which the cell dimension is approximately three times smaller
27
than the local mean free path . Though not necessarily a hard constraint, this ensures
that the simulation operates within the dilute gas limit and captures any gradients which
are present. Once the grid is constructed, it is necessary to place boundary conditions on
the appropriate cells where a boundary exists, and initialize the velocities and positions
of each particle being simulated. Often the initial conditions are such that the particles
are placed uniformly throughout the domain. The initial velocities of the molecules are
governed by the simulation itself, where the modeling of a gas with a known
ow velocity
will dier from a simulation with no bulk velocity.
One of the unique and powerful features of the DSMC algorithm is the separation of
the collision process from the physical movement of the molecules. Instead of trying to
compare both the position and velocities of two collision partners in determining whether
or not a collision occurs, the simulation instead randomizes collisions without regard
to the exact position of the molecule in the cell. The rst step in the routine will move
molecules over a small time step according to their currently assigned velocities and check
if any molecules have crossed a domain boundary. If a molecule has crossed a domain
boundary the appropriate action is taken, be it a re
ection from a surface at a specied
temperature, removal from the computation for leaving the domain, or one of any number
of other possibilities. In such cases where new parameters need to be applied, the molecule
is 'reset' by assigning a velocity from an appropriate Gaussian distribution. After this
occurs, the molecules are re-indexed to re
ect their new locations, whether they have
changed cells or just have new positions in the same cell. Finally, the collision kernel
is activated and molecular pairs are selected randomly with a collision frequency that
depends on the local mean free path. Eectively, collision pairs are chosen at random from
28
a cell with the nal result being that the collision rate predicted by theory is maintained.
For these pairs the molecular pre-collisional velocities are replaced with post-collisional
ones, and the entire process begins again in the next small time step. It is important to
note here that the post-collisional velocities assigned by the kernel are dependent on both
the pre-collisional velocities as well as the impact parameter, d! from Eq. 2.30, where it
is necessary to choose a specic approximation for the intermolecular potential to obtain
a value.
After many time steps of the above process, the macroscopic properties of the gas
are obtained. This is accomplished by sampling the velocities and cell locations of all
molecules, and creating a discretized version of the velocity distribution function in each
cell. As statistical
uctuations are extremely large for the small number of molecules
per cell, it is important that the sampling be done many times and the nal results
averaged. It is this average of the results at the individual cell level that are used to back
out everything from local temperatures and number densities, to mass and momentum
transport across cell boundaries.
As DSMC is very time consuming for both low speed
ows where the signal-to-noise
ratio is small, and for high Knudsen number
ows where the number of collisions is pro-
hibitively large, several simplied expressions of the collision term have been developed.
Of those that have been rened to a useful degree, the ones most widely used and chosen
for this work were the Bhatnagar-Gross-Krook (BGK) and Ellipsoidal-Statistical (ES)
model equations. Though such simplied models often lead to limited functionality, the
BGK model mimics the behavior of the Boltzmann equation quite well at high Knudsen
29
numbers, and though not perfect, agrees reasonably well with DSMC throughout the
range of Knudsen numbers being explored in this paper.
The foundation of the BGK model is the simplied function
@f
@t
+ v
1

df
@q
1
=
1

(f
0
f) (2.33)
where the BGK model describes the evolution of the distribution function f (q,v,t), and
collisions are modeled by the relaxation of f to the local Maxwellian equilibrium state
f
0
[34]. The similarity with the Boltzmann equation is clear, where the left side again rep-
resents the changing distribution function and the right hand side represent the collisional
contribution to that change. What is clearly dierent however is the lack of the compli-
cated collision integral, where it has been replaced with the much simpler model. First
described in 1954[6], the model has gained fairly widespread acceptance and has been
shown to be very applicable to rareed conditions in several well-regarded papers[3], [60],
[33].
The ES model employs a three-dimensional anisotropic Gaussian instead of the local
Maxwell distribution, f
0
=f
G
,
f
G
=
n
p
(2)
3
det[
ij
]
exp(
ij

0
i

0
j
);
ij
= (
1
Pr
)RT
ij
+
1 1=Pr

p
ij
(2.34)
where [] = []
1
,
~

0
=
~
~ u is thermal velocity, p
ij
is the stress tensor, and Pr is the
Prandtl number. For Pr = 1, f
G
=f
M
and Eq. (2.34) gives the original BGK model.
30
Both BGK and ES models possess the same collision invariants as the Boltzmann
equation and satisfy Boltzmann's H-Theorem expressing the increase of entropy of the
gas. Kinetic models should also reproduce the gas transport coecients - viscosity, ther-
mal conductivity and species diusivity - resulting from the Boltzmann equation; this
requirement is satised through the choice of collision frequency. The main drawback of
the BGK equation is that it fails to predict the correct Prandtl number in the hydrody-
namic limit due to the single relaxation time in the collision operator. The use of the
BGK model equation usually provides qualitatively similar, but quantitatively dierent
results, when compared to both the DSMC and the ES methods[1]. This work will, there-
fore, use the ES model introduced by Holway [21] that gives the correct Prandtl number
as the baseline.
Unlike DSMC, computational approaches to solving the BGK and ES-BGK equations
do not simulate particles statistically. Instead, the equations are solved deterministically
in a method somewhat similar to a nite-dierence approach. To accomplish this, a grid
similar to that of the DSMC calculations must be created, where it is again necessary to
have the cell size on the order of the local mean free path. Once the grid is constructed,
boundary conditions are applied to the cell walls for which conditions are known, and
initialized everywhere else at some specied start condition. Time is discretely marched
forward in steps, where the boundary conditions are propagated throughout the domain.
This is accomplished via iteration by relaxing each cell to the local Maxwellian equilibrium
state proscribed by the four "`walls"' of each cell, typically using an implicit scheme[33].
31
Chapter 3
Experimental and Computational Setup
3.1 Experimental Setup
As radiometric phenomena occur only in rareed conditions, there are only two ways to
study them experimentally. The rst of these is to build extremely small devices on the
order of nanometers and test them under atmospheric conditions. The second way is to
build a larger device and modify the background pressure such that the local Knudsen
number is large enough for the
ow to be considered transitional (i.e. Kn > .01). In this
work the latter method has been chosen, and all the experimental results that follow have
been achieved under low pressure conditions in various vacuum chambers. To study the
role of area and edge forces on the plate, three radiometer vane geometries were initially
used. Each vane consisted of a Te
on insulator sandwiched between two aluminum plates
with a resistive heater located between one of the plates and the insulator. The temper-
ature of one side of the device was maintained by varying the power input to the heater,
while the temperature of the opposite side was governed entirely by the amount of heat
lost to the environment. Each of the three pieces of the radiometer vane had a thickness
32
of 0.32cm, and when assembled using eight low-conductivity Nylon 2-56 machine screws,
yield a total device thickness of 0.95cm. The rst device used for the preliminary exper-
imental and computational comparisons was a rectangle with dimensions of 3.81cm wide
by 12.7cm tall. Figure 3.1 shows a CAD model of the device, where one side is labeled
the "`cold side"' and has a rectangular protrusion whose primary purpose was to provide
a stable attachment to the thrust measurement stand, and the other labeled the "`hot
side,"' which is in direct contact with the heater.
12.70cm
.95cm
3.81cm
Hot side
Cold side Heater
Teflon
12.70cm
.95cm
3.81cm
12.70cm
.95cm
3.81cm
Hot side
Cold side Heater
Teflon

Figure 3.1: CAD drawing of a radiometric device
The two other geometries initially tested were a larger rectangle with dimensions of
7.62cm wide by 12.7cm tall, and a circle with a diameter of 11.13cm. The dimensions of
these secondary devices were chosen such that the heated surface area of the hot sides
were identical to each other, with both having twice the area of the primary rectangular
device. Though the area ratio of the larger devices to the smaller one is the same, the
perimeter ratios are very dierent. When comparing the two rectangles, the ratio of the
larger one to that of the smaller one is 1.231. When the circle is compared to the small
33
rectangle the ratio changes to 1.058, and when the large rectangle is compared to the circle
we nd a ratio of 1.163. These dimensional choices were made with the comparison of the
three dierent devices in mind, as the historical research surrounding the radiometer has
shown that some combination of device area and device perimeter are responsible for the
force production. A comparison of the geometries and scale of all three devices is shown
in Figure 3.2.

Figure 3.2: Comparison of three experimental radiometer vanes
Another motivation for this particular conguration of radiometer vanes comes from
historical work[30] where rudimentary temperature measurements of the vanes suggested
that a signicant temperature drop occurred at the outermost edges. This same work
made it quite clear that to accurately deduce a theory for the operation of the radiometer,
it would be necessary to dene the eect of the temperature changes at the edges had.
For the sake of clarity it should be noted here that there are two gradients of interest: the
rst of these will be referred to as the radial gradient and will refer to the temperature
prole of a plate from the center to the periphery, while the second will be called the
34
axial gradient and will refer to the temperature prole along an axis normal to the face.
In an ideal experiment the axial gradient would be large and the radial gradient would
be non-existent. While a
at plate will experience more heat loss near the edges simply
due to the added surface area of the face of the edge, a radiometer adds to this heat
loss with additional convection due to thermal creep
ow driven by the temperature
gradient imposed on the gas. Interestingly, the radial gradients present in the historical
work were likely exacerbated by the specic design of the radiometer used in many of
those experiments. The majority of historical researchers used a low conductivity vane
(i.e. mica), where measurable gradients likely existed due to the marginally comparable
thermal conductivities of the vane (0.71 W/mK) and the gas (0.02 W/mK). It is for
these reasons that the particular aluminum "`sandwich"' design was chosen; not only
does the high thermal conductivity (250 W/mK) maximize the surface temperature of
the hot plate (and thus the axial temperature gradient), but it also minimizes the radial
temperature gradients near the edges of the device by conducting heat much more readily
than the gas.
To verify the benets of the sandwich design a simple thermal model was created using
COSMOS, the FEM software native to the design program SolidWorks. The model con-
sisted of the previously mentioned circular parts with widely available material properties
applied for the thermal simulation. A heat power boundary condition was applied to the
surface of the hot side nearest the insulator with an input power of 5.2 Watts distributed
over the area of the heater. An initial simulation was done with radiation being the only
mode of heat loss, and the results were compared to some initial measurements of the
35
actual device. As would be expected, this simulation produced somewhat unrealistic re-
sults with extremely high vane temperatures and a very small axial temperature gradient.
An iterative method to bring the simulation results closer to the reality of the physical
device was used in which the simulation was run several times after modifying both the
convective heat transfer coecient and the thermal conductivity of the insulator. The
results of the nal simulation are of a device which has surface temperatures and an axial
gradient very similar to that measured on the physical device, and is shown in Figure 3.3.
Though this choice of simulation leaves much to be desired from a predictive standpoint,
it is still valid as a tool to demonstrate that the radial gradient along both the hot side
(a) and the cold side (b) under the specied conditions varies less than 0.1K from the
center to the edge. It should be noted that the scales dier for each plot as the radial
temperature variation is so slight as be be unnoticeable when displayed using a range
large enough to include both the hot and cold plates.

(a) Hot side

(b) Cold side
Figure 3.3: Radial temperature prole along both faces of a radiometer
36
Each of these devices was individually mounted on a modied nano-Newton Thrust
Stand(nNTS)[24] located inside of a vacuum chamber, a schematic of which is shown in
Figure 3.4. The ner details of its operation can be found in the reference material, but a
simple explanation here seems warranted. In principle the nNTS consists of a rigid beam
attached to two torsional springs at its axis of rotation. These springs provide a restoring
torque proportional to the devices angular de
ection, such that any force applied to the
beam at some radius r will cause the beam to de
ect until the applied torque and the
restoring torque are equal in magnitude. For small angles of de
ection, this results in a
linear relationship between a force normal to the beam (and its rotational axis) and the
linear de
ection of the beam itself at the same radius in a relationship analogous to the
well known Hooke's Law:
F =kx (3.1)
Figure 3.4: Schematic of a thrust stand
For a thrust stand to be a useful tool for thrust measurement, it is then necessary to
know only two things: the rst being the position of the stand, x, and the second being
the spring constant, k. In this case the position of the stand is measured using a linear
37
variable dierential transformer, otherwise known as an LVDT. A model of the device is
shown in Figure 3.5, where the middle coil (A) is the driving coil and the outer coils (B)
are the sensing coils. During operation, the driving coil creates an alternating magnetic
eld which is passed through the movable metallic core to the sensing coils. A current
is induced in the sensing coils, and the dierential voltage produced varies linearly with
the position of the core. With the position accounted for, it then becomes necessary to
accurately know the spring constant of the test stand.

Figure 3.5: Cutaway view of a LVDT (courtesy of [61])
Measurement of the spring constant is necessarily accomplished by calibration, as even
slight changes in its value can have dramatic eects on the measured force. In practice
it has been found that the manufacturers stated spring constants are not nearly accurate
enough for scientic work, as even a 5% dierence from specication creates a 5% error
in the experimental data. The procedure for calibration is relatively straightforward and
requires only that several known forces be applied to the thrust stand while simultaneous
measurements of the total de
ection are made. In this set of experiments a pair of
electrostatic combs[45] were used to apply these forces very accurately. There are several
38
advantages to using electrostatic combs, the primary one being that force is applied
without contacting the thrust stand. This allows for in situ calibration without any
adverse eect on the actual measurement of the devices being tested. In addition, the
force produced by the combs is almost entirely insensitive to small changes in engagement.
This implies that precise knowledge of absolute position is not necessary, which both
simplies the analysis of the calibration data and reduces uncertainty in the calibration
force's value.
Figure 3.6 is a schematic showing what a pair of electrostatic combs look like. The
device consists of two comb-like pieces with prongs of identical depth. The rst comb
is mounted to the thrust stand, while the second comb is positioned such that all of
the prongs are aligned in a plane and are equidistant from each other. Using optical
positioning stages to ne tune the alignment, the initial engagement depth was set at
3mm. As shown in the reference work, this engagement minimizes sensitivity to movement
of the combs.
To use the comb as an actuator simply requires a voltage dierence to be applied
across them. In all experimental work done in this thesis the comb mounted to the
thrust stand was electrically grounded, while the comb positioned by optical stages was
connected to a power supply. This variable power supply was used in combination with
a pulse generator (DEI-4140) to apply potentials from 200 to 500 volts in increments
of 100V. For the particular set of electrostatic combs used in this experiment, the force
applied to the combs varies with the voltage using the equation
F = 1:6056E 10V
2
1:7972E 9V; (3.2)
39

Connection to
power supply Mounting holes
for positioning
stages
Ground Comb -
Attached to nNTS
Teflon Insulator
Charged Comb
Connection to
power supply Mounting holes
for positioning
stages
Ground Comb -
Attached to nNTS
Teflon Insulator
Charged Comb
Figure 3.6: Cross sectional schematic of electrostatic calibration combs
and the forces applied to the stand ranged from 6.6N to 39.2N.
A typical calibration procedure took place using an automated routine programed in
LabView. After inputting the starting and ending voltages, the voltage increment, the
number of traces at each increment (3), and the trace length (300s) into the program,
the routine was started. At a precise time (45s after starting) the pulse generator would
trigger and voltage would be applied for 150s. Eectively just a simple looping structure,
the program would run all traces at a given voltage and then proceed to the next higher
voltage. During a given trace the software would log both LVDT output voltage and
pulse generator output voltage at 60Hz. As the 60Hz spectrum is notoriously noisy, signal
ltering was necessary. The time resolution of the data has no bearing on the results being
observed, so a simple moving window averaging method was applied. Beyond ltering the
noise, the analysis of the calibration data was straightforward and was done in Microsoft
Excel. The process was as follows: 1) The rst 15 seconds of a trace was averaged and
40
subtracted from the entire data set, which brought the entire trace to
uctuate around
zero and dened the x-axis as zero de
ection. 2) The last 15 seconds of the trace was
averaged and the slope between this average and the previous was calculated. 3) This
slope was the thermal drift of the thrust stand, and its eects were removed by multiplying
each point by the slope and subtracting it from its measured value. 4) The nal values
of each point were plotted on a corrected plot, again with the x-axis being time and the
y-axis being de
ection. 5) The total de
ection of the device was calculated by averaging
the voltage over a 15 second window centered about 150 seconds.
An example of a calibration trace at 500V is shown in Figure 3.7(a), where time is
plotted on the x-axis and de
ection in Volts on the y-axis. Also shown(b) are the combined
results of the analysis of all of the calibration traces, with the line t using a least squares
regression and forced through zero. Here, each data point represents the average of three
measurements, where the standard deviation of each is less than one percent. The slope of
this line represents the eective spring constantk
eff
, with the peculiar units of N/V. The
noise shown in the raw signal is a combination of both electronic noise and small, physical
vibrations of the thrust stand. Though seemingly large compared to the measured signal,
it can be seen in both the statistics of the calibration data, as well as the linearity of the
points, that the ltering method used works extremely well.
When calibrated using the above procedure, the nNTS provides very accurate and re-
peatable data with typical force resolution of approximately 0.1N and statistical scatter
of about 1%. For the preliminary experiment, the experimental error based on standard
deviation ranges from a few percent at the lowest pressures to less than 1% through
most of the curve. However, due to the normalization by experimental temperature
41
(a) Corrected calibration trace (500V)
y = 2.524E-06x
0.00E+00
5.00E-06
1.00E-05
1.50E-05
2.00E-05
2.50E-05
3.00E-05
3.50E-05
4.00E-05
4.50E-05
0 2 4 6 8 10 12 14 16
Deflection, mV
Force, N

(b) Summary of calibration traces
Figure 3.7: Calibration results
measurements and the small uncertainty of the calibration method, the total absolute
experimental uncertainty is 4%. Day-to-day variation of multiple data sets has been
observed to be 1%.
The experimental data was obtained for each radiometric device by evacuating the
vacuum chamber to a base pressure below 10
4
Pa. This low pressure was required to
minimize the impact of the background gas to a level low enough as to be inconsequential
to the measurements being made. These pressures were obtained in two dierent stainless
42
steel vacuum chambers, the rst of which was 0.4m in diameter and the second was 3m in
diameter. As the experiment progressed multiple congurations of these chambers were
used, and ultimately it was discovered that the simplest and most repeatable way to take
the data was to do all of it in the same chamber. As there were signicant modications
of the experiment made in moving from the small chamber to the large one, the changes
will be noted in this section with the major explanations of why these changes were made
in the sections to follow.
The initial experiments were done in a 0.4m diameter stainless steel vacuum chamber
with the nNTS contained entirely inside the chamber along its axis. A layout of the
chamber is shown in Figure 3.8, where it can be seen that there was a long tubular
section connected to an ISO-400 cross. At the far end of the cross there was a 90 degree
elbow attached to a mechanical gate valve, below which sat a turbomolecular pump. At
the pump end of the chamber, near the center of the cross, the radiometer vane was
attached to the test stand. At the opposite end of the tube is where both the calibration
combs and the LVDT were mounted. A gas inlet line with a mechanical toggle valve
was used to control the presence of a background gas, while an in-line needle valve was
utilized to control pressure.
While the evacuation of the chamber was taking place, a constant voltage was applied
to the heater. This resulted in the main radiometer surfaces reaching temperatures of
approximately 415K (hot) and 365K (cold), though the exact values
uctuated depend-
ing on which device was being measured. The initial warming of the device to a steady
temperature prole took approximately two hours once the chamber was evacuated below
43

Gate Valve
Turbo
Pump
Optical Bench
Gas Inlet
Thrust Stand
Radiometer
Pressure
Sensor
Gate Valve
Turbo
Pump
Optical Bench
Gas Inlet
Thrust Stand
Radiometer
Pressure
Sensor
Figure 3.8: Initial conguration of 0.4m chamber
10
2
Pa, and at the end of the warming process the daily calibration procedure previ-
ously described was initiated. Once this nished approximately an hour later, the actual
experimental data sets would be run.
In a manner similar to calibration, the rst 30 seconds of an experimental trace would
be taken with nothing happening. This established a zero condition for the data set, which
ensured that measurements were always compared to the actual initial conditions present
in the vacuum chamber. At this time the background pressure would be measured using a
0.01mTorr Baratron, and the temperature on both the hot and cold side of the radiometer
logged using J-type thermocouples paired with a hand-held, temperature compensated
meter. After 30 seconds, the background pressure inside the chamber was varied by

ooding the chamber with a specic gas over a range of pressures from 0.1 Pa to 6 Pa.
Argon, helium, and air were all utilized with the additional limited use of xenon. The
system would typically reach quasi-equilibrium values at around 90 seconds, and the
pressure and temperature would again be logged at 120 seconds. After 150 seconds of

ooding the chamber the inlet would be closed and the remaining background gas would
44
be pumped out, with the system returning to equilibrium with the original conditions by
the end of the 300 second trace.
Flooding was accomplished initially by sealing the gate valve and letting a limited
amount of the test gas into the chamber. This method immediately ran into problems
as it was dicult to obtain small chamber pressures, and even harder to take multiple
points at the same pressure. A dierent method was tried in which the gate valve would
be left open slightly and a small amount of gas would continually
ow into, and out of,
the chamber. When the
ow rate was set by the needle valve, measurements taken in
this manner were much more consistent and compared well with the data taken under
stagnant conditions. A typical trace using the variable
ow rate method mentioned above
is shown in Figure 3.9, where a ltering method identical to that used for the calibration
data is also employed.
Figure 3.9: Representative trace of ltered data, background pressure 0.52 Pa.
45
Using this experimental chamber setup, data was only taken for the small rectangle
as it was quickly discovered that this was an unsuitable conguration for several reasons.
The primary problem was that the location of the gas inlet created an extremely noisy
environment at higher background pressures for the nNTS, where the incoming gas would
impinge on the thrust stand arm and likely created experimentally signicant
ow elds.
Another problem was due to the direct physical connection of the radiometric device to
the nNTS, where large heat conductivity to the thrust stand caused a secondary eect
which increased the force production. With these notable limitations, the thrust stand
and test devices were moved to the much larger 3m chamber.
The layout that was nally decided upon was selected not just for its experimental
value, but also due to the ease with which it could be computationally modeled. In the
3m chamber the nNTS was mounted radially, instead of the previous axial setup in the
small chamber. This was possible due to the much larger dimension of the 3m chamber,
and was extremely practical as it also allowed for direct comparison with axis-symmetric
simulations (at least for the circular geometry). As shown in Figure 3.10, the stand was
mounted in the middle of the large chamber 1.5m from the door and 4.5m from the
rear wall. The radiometer itself was positioned such that the center of the device was
aligned with the axis of the chamber, with the hot side facing the rear wall and the cold
side facing the front door. The nNTS extended radially outward from the center of the
chamber, with the calibration combs and position-sensing LVDT mounted very close to
the chamber wall.
In this conguration data was taken for all three devices shown above using research
grade argon in a manner identical to that used in the small chamber. Due to limitations
46
Figure 3.10: Initial conguration of 3m chamber.
in the pumping system attached to the larger chamber, the lowest background pressure
achieved was 10
3
Pa. Similar problems limited the maximum pressure attained to ap-
proximately 2Pa, so the experimental background pressure ranged from 0.1Pa to 1.6Pa.
As the location of the gas inlet had previously created a problem for thrust measurement,
a dierent location for the nNTS was chosen in the large chamber. In this setup, the gas
inlet was located in a baed region near the back end of the chamber, while the thrust
stand itself was located several meters away near the front of the chamber. To verify that
the measurements were not aected by the gas being bled into the chamber, a simple
test was performed. With the heater to the radiometer turned o, gas was allowed into
the chamber at various
ow rates as would be done in a normal test. The de
ection
47
of the thrust stand was logged, and the nal results were analyzed. A typical trace of
the result is shown in Figure 3.11, were movement is indeed detected. Fortunately, the
motion was limited to the regions of the trace corresponding to the opening and closing
of the valve, with the
ow itself having no in
uence on the stand once the pressure in
the chamber equilibrated. Looking at the trace, it is easy to see why this is so: rst
the force measurement spikes as the pressure wave passes by the stand and the chamber
equilibrates. As the chamber comes to equilibrium the measurement returns to normal
(i.e. no de
ection of the stand). Finally, the valve is shut and the pressure gradient
reverses, causing the stand to de
ect in the opposite direction as the background gas is
pumped from the chamber.
Figure 3.11: Impact of gas
ow on thrust measurement.
During the initial data analysis, a signicant decrease in the force measurements was
discovered to occur in the large chamber at higher pressures (as compared to the previous
small chamber data). A signicant deviation of the experiment from the computational
48
results was also noted. With this in mind, it was decided that an experimental comparison
between the dierent chambers was necessary, and that a dierent attachment mechanism
was required to improve the comparison with simulation.
To examine the eect the size of the chamber had on force production the small
chamber data was obtained a second time, this time without impact from the inlet
ow.
In this set of experiments, the initial ISO 400 chamber was disassembled and only the
cylindrical portion of the chamber was used. The remaining tube was positioned inside
the 3m chamber with its axis centered on the arm of the thrust stand. The midplane of
the tube was aligned with the center of the radiometer vane, and aluminum foil was used
to cover the open ends while leaving enough of a hole at one end for the thrust stand arm
to pass through without contact. This conguration is shown in Figure 3.12. To verify
that the pressure measured in the large chamber was identical to that inside the tube, a
dierential Baratron rated for 2.700.01 Pa was used to measure the pressure dierence
across the wall of the tube. Through a wide range of pressures, the Baratron recorded no
measurable dierence. The temperature of the tube wall was also monitored, with the
tube wall increasing slightly above background gas temperature by one to two degrees
Celsius. Data was taken for this conguration in a manner identical to that of the large
chamber. All three devices were tested across the same pressure ranges, with the same
pumping limitations as previously mentioned.
With the eect of chamber size quantied, two new types of attachments were pro-
posed and built. Both attachments are shown in Figure 3.13, where one was designed to
minimize the impact on the radiometer edge with a 2-56 threaded rod used for mounting,
and the other designed to completely eliminate any impact on the edge by mounting
49
Figure 3.12: Conguration of ISO 400 tube in 3m chamber.
directly to the center of the cold side. Both designs relied on 1/4 inch tubing as a sup-
porting framework which xed the radiometer to the nNTS. Data for both new designs
was taken using the same pressure ranges mentioned previously, and it was discovered
that mechanism (a) was the better candidate for continued research.
With a new method of attachment selected, the previous devices were modied by
cutting o the large attachment arms and tapping a small hole in the Te
on insulator into
which the threaded rod would be inserted. The 1/4inch tube was attached to the thrust
stand, and the experiment proceeded just as before with the pressure being varied over
the previous range using argon as the background gas. In addition to the three devices
used previously, a small circular device was also included. Identical in thickness to the
50

(a) Side mounted radiometer

(b) Back mounted radiometer
Figure 3.13: Attachment mechanisms designed to minimize impact
other devices, it had a diameter of 8.59cm. Figure 3.14 shows the modied devices with
the addition of the smaller circle for visual comparison. Due to the loss of a conductive
heat path through the attachment mechanism, the axial gradient across the vanes were
noticeably reduced for each radiometer. Typical temperatures for the small rectangular
device were 429K (hot) and 407K (cold), though dierent devices still had slightly dierent
temperatures. As a nal experimental comparison, the large circular device was tested
using a variety of background gases including: argon, carbon dioxide, helium, nitrogen,
and xenon.
51
Figure 3.14: Comparison of nal four experimental radiometer vanes
The experimentally measured force was normalized by the temperature dierence
between the hot and cold plates, T, for the purpose of comparing results for dierent
pressures and gases. Verication of the validity of the experimental normalization method
is demonstrated in Figure 3.15, where exceptional linearity with temperature dierence
is observed. Three pressures are plotted at various temperature dierences, where each
pressure corresponds to a dierent
ow regime. It is clearly shown that the force produced
by the radiometer remains linear with T regardless of the pressure (within the tested
range)at which the data is taken.
52
0.00E+00
2.00E-06
4.00E-06
6.00E-06
8.00E-06
1.00E-05
1.20E-05
0 4 8 12 16 20
ΔT (K)
Force, N
.56 Pa
1.36 Pa
2.69 Pa

Figure 3.15: Changing force as a function of temperature dierence across surfaces
3.2 Computational Setup
For the initial comparison with experimental results the direct simulation Monte Carlo
computational tool, SMILE[23], was used by Sergey Gimelshein in all DSMC computa-
tions presented in this work. The variable hard sphere model with parameters put forth
by G.A. Bird[8] was used for the intermolecular potential, and the Larsen-Borgnakke
model with variable rotational relaxation number used for translational-internal energy
transfer. The gas-surface model was assumed to be diuse with full energy and momen-
tum accommodation. A two-dimensional module of SMILE was used in the bulk of this
work because three-dimensional modeling is prohibitively expensive for most pressures
under consideration. The computations were conducted for two plates with dimensions
0.95x3.81cm and 0.95x7.62cm immersed in an initially uniform stagnant gas in a chamber.
The third (Z axis) dimension of 12.7cm was assumed when calculating forces consistent
with the actual size of the plate in the present experimental study (for the rectangular
devices). The main surfaces of the plate were set to 410K (cold) and 450K (hot), and
the sidewalls were 430K. The chamber wall temperature was assumed to be constant at
53
300K. The computations were performed for chamber pressures ranging 0.006Pa to 6Pa
and chamber sizes from 0.45cm to 1.8m using three gases: helium, nitrogen, and argon.
A nite volume solver SMOKE, developed by N. Gimelshein[55] based on the ES-BGK
numerical scheme in [33], was used in the remainder of the
oweld computations. Again,
the gas-surface model was assumed to be diuse with full energy and momentum accom-
modation. The majority of the
oweld results were solved using two-dimensional
at
plates, while any direct comparison with experimental results required an axis-symmetric
calculation. Three-dimensional calculations were avoided due to the complexity and size
of the domain and the corresponding amount of computational time required to implement
them. With the expectation that experimental conrmation may be desired in the future,
the vane dimensions of the computational study were chosen to re
ect similar dimensions
to those of the experimental work. This considered, the computations were conducted for
multiple vanes with dimensions 1.0x10.0cm immersed in an initially uniform stagnant gas
in a chamber 0.5x0.8m. It should be noted that one wall of the chamber was set with a
specular boundary, such that the chamber (and enclosed vanes) were re
ected about this
wall as if it were an axis of symmetry. This eectively doubled the size of the chamber
and the number of vanes, such that the simulation modeled both the described domain
and its mirror image. In this work, results will be shown for two, three, ve, and nine
vane cases. An additional case was added for comparison, in which a single vane with
dimensions of 1.0x55.0cm was placed at the center of an identical chamber. This allowed
for numerous comparisons between cases of both varying vane separation and numbers
of devices, and provided meaningful insight into how multiple vanes impact one another.
The main surfaces of the plate were again set to 410K (cold) and 450K (hot), with the
54
sidewalls at 430K. The chamber wall temperature was maintained at a constant 300K.
The computations were initially be performed for chamber pressures over a wide range of
pressures to determine the pressure at which the force peaks. Once the peak was located,
computations were made over a suitable range of pressures (0.0063Pa to 0.5Pa) such that
the shape of the force curve is adequately determined. Due to the extremely large amount
of computer time required for these runs, the full range of pressures (0.0063Pa to were
only run for the fastest cases (two and three vane). For the more complicated cases only
the lower pressures were simulated, and for the nine vane case only the 0.5Pa case was
run.
As one of the stated goals of this work was to obtain meaningful values of the accom-
modation coecient, a more suitable method of simulation was required. Ultimately it
was decided that comparison of an axis-symmetric calculation with the circular experi-
mental device would yield the most accurate results, and a study was undertaken. For
this nal comparison a hybrid approach was used, wherein both DSMC and ES-BGK were
utilized. To accomplish this, a very large domain was created with the dimensions of the
large experimental vacuum chamber(3.0 x 3.0 m). The vane was located at the center of
the chamber as in the experiment, and the ES-BGK solver was run for several thousand
time steps until steady-state was reached. Boundary conditions were obtained from this
simulation for a smaller domain of 0.6 m, and this information was passed on as the initial
conditions in a DSMC simulation. These DSMC results for the net force produced by the
radiometer are directly compared with those of the large circular experimental vane.
55
Chapter 4
Gas Species: The Eects of Collisional Cross Section and
Accommodation Coecient
The forces from gas on hot and cold surfaces were computed for a 2-D rectangular plate,
and the dierence between the two (the net force) was analyzed for three carrier gases,
helium, argon, and nitrogen. The net force for these gases as a function of pressure is
given in Fig. 4.1.

Figure 4.1: Computed force for various gases
56
It is seen that for every value of pressure under consideration the force is larger for
the gas with longest mean free path, helium; it is minimum for nitrogen, which has
the smallest mean free path. This is related to the fact that the force per molecule
is maximum in the free molecular regime, and it decreases as soon as molecules start
colliding with each other. Moreover, it tends to zero as the Knudsen number decreases,
where continuum gas eects begin to dominate over those created in the transition regime
(thermal creep, transpiration, etc.).
The dependence of force on the mean free path of the gas is illustrated in Fig. 4.2,
where the net force normalized by the force on the hot plate is shown as function of
the Knudsen number calculated near the hot surface. There does not appear to be an
impact of internal degrees of freedom for the temperatures under consideration. It is also
important to note that the normalized force for three gases is within the error bars of the
computational results that are estimated to be about 5% based on the statistical scatter
of the surface properties.
Experimental results for the smallest rectangular device in the original 0.4m cham-
ber conguration are shown in Fig. 4.3, where they are normalized by the temperature
dierence across the plates. Normalization was necessary to account for dierences in
measured force due to slight changes in the magnitude of the gradient over the course of
the experiment.
A qualitative comparison of Fig. 4.1 and Fig. 4.3 displays their similar behavior in
both trend and peak location. It is readily noted that helium produces a much smaller
force proportionally in the experiment. It is believed this is caused in part by the low
energy accommodation of helium. After comparing the low pressure results from these
57

Figure 4.2: Normalized force vs Knudsen of various gases
initial experiments with calculations in the collisionless regime, it was discovered that
undesirable heat transfer to the nNTS was decreasing the experimentally measured force.
This reduction was due to a temperature gradient opposite that of the desired experimen-
tal one, which acted as a 2nd radiometer eectively in opposition the the experimental
device. In all following experimental plots, this situation was remedied by taking the ex-
perimental data in a dierent conguration. The second round of experimental work was
notably improved with the much smaller attachment arm, and the results for a variety of
gases in the large chamber are shown in Fig. 4.4. Here, the large circular vane was used
for all tests cases.
In this plot it can clearly be seen that helium produces the largest forces of any
of the gas species, with total force production again dropping o as the collision cross
section increases. It is also seen, however, that helium produces signicantly lower forces
at the lowest pressures ( 0.2 Pa). This result can largely be explained by the eect
58
0.00E+00
2.00E-07
4.00E-07
6.00E-07
8.00E-07
1.00E-06
1.20E-06
0 1 2 3 4 5 6
Pressure (Pa)
Force/ΔT, N/K
Argon
Air
Helium
Xenon

Figure 4.3: Experimental results for various gases in original small chamber
of the accommodation coecient, and oers further proof that the large discrepancies
between calculation and experimentation can be attributed to the non-unity values in
the experiments. Similar results in which micro-cantilevers experience meaningful force
reductions have also been presented by Gotsmann and D urig in 2005[17].
Finally, Table 4.1 compares selected experimental results from the previous gure
with those obtained from the nal axissymmetric hybrid DSMC/BGK computations in
a 3.0 x 3.0 m chamber for argon, helium, and xenon. Here, the results are presented
in units of Newtons, where the experimental values were multiplied by the temperature
dierence used in the calculations (25 K). It is clearly seen that all of the experimental
results fall below their computational counterparts, with the dierence between them
being most notable using helium. A ratio is taken by dividing the experimental results
by the computed ones, and the results are averaged at the bottom of each data set. This
average is used to calculate the accommodation coecient from eq. 2.12, and the
calculated values appear below the ratio averages.
59
Figure 4.4: Experimental results for various gases in large chamber
To test the validity of the assumptions used in the derivation of , a second hybrid
computation was run using the calculated value = 0.50 in the code. Results of that
test are shown in Fig. 4.5, where they are compared with the original computed values
( = 1) as well as with experimental values. Fairly good agreement is found in comparison
between the results, where it is observed that the measured and computed forces
uctuate
about each other.
In Table 4.2 a comparison of the calculated values is made with historical ones from
Table 1.1, as well as values recently presented by M.A. Gallis[53], [16]. Here, agreement
with both historical and recent values is less than ideal, but is consistent with the historical
trend of widely diering values and still exhibits reasonable proximity with both Ar and
He. This variation in coecient is seen especially for the nitrogen values, where a range
from 0.76 to 0.95 is presented. Nitrogen values were not calculated computationally due
to the internal energy modes of the diatomic molecule. The calculated value of for xenon
on aluminum is also presented; unfortunately there is little published data to compare it
60
Table 4.1: Comparison of experiment and computation for Ar, He, and Xe
with. It should be noted that the widely published trend[48] of increased accommodation
with increasing molecular mass is found to hold true here as well.
Table 4.2: Comparison of computed values of accommodation coecient with various
sources
61
Figure 4.5: Comparison of experimental and axissymmetric computational helium results
for 3.0m chamber
62
Chapter 5
Geometry Considerations: Area, Perimeter, and
Attachment Eects
To examine the contribution of the edge eects to the net force, and analyze the relative
importance of the forces on the edges of plate versus the area forces, the preliminary
2-D DSMC computations were performed in helium, for two geometric congurations, (i)
a solid 3.9cm plate, and (ii) a 3.9cm plate that has ten 1mm holes. Consider rst the
surface pressure distribution over the cold and hot sides of the solid plate, shown in Fig.
5.1 for a chamber pressure of 2Pa.
The pressure has a minimum near the center of the plate both for the hot and cold
sides. The values near the edges are visibly larger than that near the center. The net
force, however, is produced by the dierence of pressure forces on the hot and cold sides of
the plate, and this dierence is nearly constant for dierent locations along the plate. This
is an indication that the area contribution is very important to the net force. Additional
conrmation is given in Table 5.1. The force on a hollow plate is generally lower than
that on a solid plate, with the only exception being for the 6Pa case where it is within
the uncertainty of the computations.
63

Figure 5.1: Pressure on the main surfaces of the radiometer
The
ow eld structure typical for the transitional regime around the solid plate is
shown in Fig. 5.2, where the translational temperature and streamlines are shown for he-
lium at a chamber pressure of 2Pa. Here, the computation domain measures 0.45x0.45m,
and is bounded by 300K diusely re
ecting walls. The hot side of the vane is on the
left hand side of the plate shown in the gure. The four vortices are created by the
temperature gradients in the gas, where two are formed on each side of the plate. The
circulations on the hot side of the plate are noticeably stronger than on the cold side. On
the hot side, the maximum
ow speed near the line y = 0 is about 5m/s. Comparatively,
the colder vortex is much slower and has a
ow speed of approximately 2m/s. Note that
the vortex structure is similar to that calculated by Ota [35], but qualitatively diers
from the
ow structure given by Kennard [25].
64
Pressure: 0.006 Pa
Solid Plate Hollow plate
Hot surface 2.88430E-5 2.15860E-5
Cold Surface -2.81203E-5 2.15860E-5
Net Force 7.22630E-7 5.49579E-7
Pressure: 2 Pa
Solid Plate Hollow plate
Hot surface 1.01827E-2 6.85919E-3
Cold Surface -1.00775E-2 -6.77046E-3
Net Force 1.05428E-4 8.85723E-5
Pressure: 6.092 Pa
Solid Plate Hollow plate
Hot surface 3.00330E-1 2.04556E-2
Cold Surface 2.99301E-2 -2.03487E-2
Net Force 1.02870E-4 1.06145E-4

Table 5.1: Comparison of solid and hollow radiometer vanes
To further explore the dependence of the radiometric force on area and perimeter,
pressure distributions on the plate surface were calculated using a secondary kinetic
approach across a range of pressures. The dierence in pressure between the hot and
cold sides is obtained through the solution of the ES-BGK equation, and is presented
in Figure 5.3 for argon gas. Only half of the plate is shown due to the symmetry of
the
ow, and Y = 0 corresponds to the center of the plate. The presented pressure
dierence essentially determines the magnitude of the radiometric force. Note that the
DSMC results are close to ES-BGK, although they have noticeable statistical scatter.
As a result, the DSMC results are not shown here. For the lowest pressure of 0.305 Pa,
the pressure dierence, as well as individual distributions over the cold and hot surfaces,
is nearly
at. This indicates that free-molecular eects dominate at this pressure, even
though based on the Knudsen number (Kn 0:5) the system is far from free-molecular. As
pressure increases, pressure dierence near the edges becomes larger than at the center,
65
Figure 5.2: DSMC (top) and ES-BGK (bottom) streamlines and temperature eld in
helium.
thus showing that the \collisional" thermal transpiration forces become noticeable. It is
important to note that the pressure dierence is still relatively
at near 1.2 Pa where the
force is maximum, and that the impact of the edges propagates much further than the
single mean free path suggested by Einstein and others[13],[42]. This essentially increases
the importance of the vane area, and suggests that increasing the open area of a vane to
maximize perimeter is not an ideal way to increase the total force produced by a device.
The preliminary experimental results for nitrogen in the small chamber are shown in
Fig. 5.4. It is readily observed in the low pressure region that the circular plate and
large rectangular plate collapse onto each other and that their values are very close to
twice that of the smaller plate. This is expected, as free molecular theory shows the force
to be entirely dependent on area. As the
ow transitions from the collisionless regime,
the picture becomes distinctly more clouded. While it is readily observed that the plates
with larger area produce more force at their respective peaks, the force/area ratio does
66
Y, m
ΔF/S, Pa
-0.015 -0.01 -0.005
0.005
0.01
0.015
0.02
0.305 Pa
1.219 Pa
2 Pa
6.092 Pa
Figure 5.3: Computed pressure dierence along the plate
not hold throughout the pressure range. In fact,as the background pressure increases and
the larger device begins transitioning, it can be seen that the smaller device has not yet
peaked.
When comparing the peaks of the large and small rectangles, it is found that the small
plate creates 65% of the force of the larger one. Interestingly, this is almost precisely half
way between the area ratio (50%) and the perimeter ratio (81%). Were that the only
data presented, it would be tempting to presume that the area dependence of the force
was being superseeded by the perimeter dependence predicted by Einstein. This does
not entirely appear to be the case however, as the peak force created by the circle is
decidedly larger by several percent. It seems quite contradictory to the accepted theory
that a device with 14% less perimeter somehow creates a larger total force without an
equivalent increase in heated surface. This increase in force is possibly explained by the
67
0.00E+00
2.00E-07
4.00E-07
6.00E-07
8.00E-07
1.00E-06
1.20E-06
1.40E-06
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Pressure (Pa)
Force/ΔT, N/K
Circular Plate
Large Plate
Small Plate

Figure 5.4: Normalized force for dierent device geometries in the 0.4m chamber
symmetry of the circle, where the vortices shown in Chapter 7 may negatively interact
at the corners of the large rectangle, or more readily with the attachment mechanism.
The numerical predictions obtained for two dierent plate sizes are shown in Fig. 5.5
and are qualitatively similar to the above measurements. In the free molecular regime the
larger plate produces a proportionally larger force. The ratio of the small plate force to the
large plate force increases with pressure and reaches 0.7 in the range of pressures where
the force is maximum. This furthers the case that the area is still a large contributor to
the peak force, where it must be noted that the perimeter ratio of the 2-D case is unity.
Comparison of the DSMC results and experimental measurements of the net force on
the plate in nitrogen is shown in Fig. 5.6 for dierent pressures. The force is normalized
by the temperature dierence between the cold and hot surfaces, since the actual values
of the temperature somewhat varied in the experiment. The comparison shows that
both measurements and DSMC predict pressure maximum around 1 Pa. The numerical
68

Figure 5.5: Numerical results for two dierent plates
results noticeably over predict the measurements, with the dierence attributed primarily
to the three dimensionality of the experiment, the impact of the mounting arm, and an
incomplete energy accommodation which was not accounted for in DSMC.
It must be noted here that the large discrepancy between the experimental results
and the computation at higher pressures was a cause for concern. Beyond the somewhat
expected disagreement with the preliminary computed results, there were several pieces
of evidence suggesting that the attachment mechanism was likely reducing the force by
at least 10 percent. The rst of these was an early
oweld plot (see Chapter 7, which
showed that the
ow structure near the force peak was surprisingly complex. Knowing
that the attachment was blocking at least 10 percent of the
ow due to its large size,
it became an immediate candidate for the cause of the discrepancy. Further proof came
from an early experiment done entirely in the large chamber, when compared to an
axisymmetric calculation done at a higher pressure the results were still approximately
69

Figure 5.6: Comparison of experimental and computational results
20 percent dierent, with the calculations again over predicting experiment. It was at this
point in the experimental work when it became apparent that the attachment mechanism
was going to have to be modied, and where the two candidates mentioned in Section 3.1
were fabricated and tested.
As noted, the rst new attachment mechanism still protruded radially from the side
of the radiometer vanes, while the second attached directly to the center of the cold vane
along the axis of the device. To test which device minimized the eect of the
oweld,
each attachment was individually mounted to the large circular vane, and then fastened
to the nNTS. The results for argon are presented in Figure 5.7, where large increases in
the maximum force produced are observed. For the small side mounted attachment, an
increase in force of approximately 20 percent was measured, with a much larger increase
being observed for the center mounted one.
70
0.00E+00
2.00E-07
4.00E-07
6.00E-07
8.00E-07
1.00E-06
1.20E-06
1.40E-06
1.60E-06
0.000 0.200 0.400 0.600 0.800 1.000
Pressure (Pa)
Force / ΔT (N/K)
Center Attachment
Side Attachment
Wide Attachment
FM Solution

Figure 5.7: Comparison of experimental results for 3 attachment mechanisms.
Looking at the low pressure data for these devices it is observed that the small side
attachment and the wide attachment agree quite well, with both data sets being slightly
below what free molecule theory would predict. As the wide attachment mechanism had
no axial gradient across it, this agreement is precisely what would be expected with some
leeway allowed for the uncertainty of the data as well as a non-unity accommodation
coecient. The center mounted attachment, however, created forces that were several
percent above those predicted by free molecular theory. Initially thought to be an er-
roneous data set, several other tests conrmed that the results were indeed repeatable.
After searching for the cause of such an odd result, insight was again taken from the

oweld plots. The added force can readily be explained by the interaction of the
ow
as it streams away from the center of the cold plate, not only colliding with the axial
mounting arm, but also with the transverse section attached to the thrust stand. As the
increase persists even in the most rareed environments tested, the eect is likely due
71
to what shall be called "`momentum recovery"'. In this scenario molecules re
ecting o
of the cold side attain average velocities which are higher than those of the surrounding
gas. As some fraction of these molecules will collide with the attachment mechanism,
the momentum of the colliding molecules will transferred to the attachment arm. This
recovered momentum will actually add to the force produced by the radiometer, as the
collisions arrive preferentially from the direction of the axial gradient (ie, the direction of
the force).
Observing that the attachment arm is such a small fraction of the total space into
which molecules may re
ect, it may seem as if this would be a trivial amount of momentum
exchange, and not nearly enough to contribute meaningfully to the total force production.
This is misleading however, in that the force produced by the radiometer is an extremely
small fraction of the total force produced by each side of the vane. In fact, the force
produced by the radiometer is only about 6 percent of the total force produced (at peak
force) by each plate, where the measurable force is ultimately the dierence in pressure
between the hot and cold sides. When viewed in this light, it means that even a small
recovery of momentum coming o of the cold side of 1 percent, is the equivalent of adding
about 17 percent to the total output of the radiometer.
At the higher pressures the dierences between attachment mechanisms is equally
striking. Not only did both of the new attachments increase the force production of the
radiometer vane, but the center mounted attachment surprisingly showed an increase of
well over 50 percent. For the side mounted attachment the force gain appears appropriate
for two reasons: rst, it maintains realistic results in the low pressure region where the
attachment mechanism ideally would have no eect, and second, the removal of the large
72
attachment blocking 10 percent of the
oweld could easily restore an equal 10 percent
of the force and have the added benet of creating almost ideal circular symmetry. The
very large force increase for the center mounted attachment also seems warranted, as
the
oweld plots indicate that the
ow near the center part of the vanes is hottest and
fastest. When coupled with the idea of momentum recovery, it is entirely plausible that
much larger forces could be obtained. From an engineering standpoint, this conguration
is extremely practical as it appears to recover wasted energy. Indeed, the ideal case of a
radiometer as put forth by Einstein is one in which only one side of the vane is heated
and the opposing side never increases above the ambient temperature of the container.
This makes perfect physical sense, as any energy being put into the gas on the cold side
of the vane is energy which is not only wasted, but also counter-productive in that it
ends up reducing the total motive force of the radiometer. With regard for the future
design of such devices, it is extremely important to note that not just the axial gradient
is important for force production, but also that recovering wasted energy from the cold
side, or not wasting it in the rst place, are extremely important to ecient operation.
From the above data it was decided that the side attachment would be used for the
remainder of the experimental work. Further, this side conguration is much closer to
the newest computer simulation model using an axissymmetric code, and interestingly
removed the vast majority of the previous discrepancy between experimental and com-
putational results. Though the center mounted attachment produces much larger forces,
they are unfortunately produced under much more complicated circumstances. Though
73
these larger forces would be experimentally useful in that they would reduce the uncer-
tainty in the measurements, the added complication of trying to understand and model
the extremely asymmetric
oweld makes this attachment mechanism less than ideal.
The results of the experimental study of the newer radiometer shapes (vanes without
the wide attachment) mounted to the small side attachment are summarized in Fig. 5.8,
where the small circle is shown in place of the small rectangle. As the results from 5.4
demonstrated that the comparison of the large rectangle and large circular geometries
did not show the expected perimeter eect, it was believed that a direct comparison of
two circles of dierent sizes would give a better understanding of how a specic geometry
transitions from free molecule to continuum
ow. Without the contribution of the wide
attachment clouding the results, it is observed that the large circular device and large
rectangular device of equal areas still do not perform in proportion to their perimeters,
as they produce almost equal forces throughout the pressure range tested.
Pressure, Pa
Force/ΔT, N/K
0.2 0.4 0.6 0.8 1
2E-07
4E-07
6E-07
8E-07
Small Circle
Large circle
Rectangle
Figure 5.8: Measured force on three dierent plates in argon
74
These results are also presented as force ratios as a function of pressure in Fig. 5.9,
where it is clearly seen in the low pressure region that the force is proportional to the
plate area. This is decidedly consistent with predictions made by free molecular theory.
As the
ow transitions from the collisionless regime, the picture becomes distinctly more
complex. While it is readily observed that the plates with larger area produce more force
at their respective peaks, the force/area ratio still does not hold. When comparing the
peaks of the large and small circular plates, it is found that small plate creates 72% of
the force of the larger one. Interestingly, this remains somewhere between the area ratio
(60%) and the perimeter ratio (77%), as was seen in the early experiments. Again, it
is tempting to presume that the perimeter force is beginning to dominate, and again
this is not wholly the case. The uncertainty continues to stem from comparison of the
larger vane data, where the force produced by the circle is nearly identical to that of the
large rectangle. This unexpected result may again be explained by more ecient pressure
redistribution in front of the circular plate.
In summary, it is worth noting that radiometric force production by a solid, single
vane is a multi-faceted eect. At extremely low pressures (high Kn numbers) the device
operates as if the gas has no intermolecular collisions, and force production is completely
governed by the area and temperature dierence of the device. At very high pressures
(low Kn numbers) the device behaves such that only a small area near the perimeter of
the device is active, though Fig. 5.3 clearly shows that this is much larger than one free
path wide.
75
Figure 5.9: Comparison of force ratios for 2 geometry pairs
76
Chapter 6
Volume Considerations: Chamber Eects
The impact of the chamber walls has been studied for a helium
ow at 2Pa with the
chamber size varied from below 0.2m to about 1.8m with the boundary conditions on
the plate and chamber walls unchanged. The analysis of the
ow eld shows that even
when the chamber is about a hundred times larger than the heated plate, there is still
pronounced eect of the location of the chamber walls on the
ow properties not only near
the chamber walls, but also near the plate. This is illustrated in Fig. 6.1 where the tem-
perature proles are shown for dierent chamber sizes in the cross section perpendicular
to the plate and coming through its center. The plate center is at X=0.
Generally, the increase in the chamber size results in decrease of temperature and
pressure gradients near the plate. As a result, the net force exerted on the plate decreases
for larger chamber sizes, as shown in Fig. 6.2. It is also shown that a rapid increase in
force occurs as the chamber size decreases, and it is expected to further increase until the
free molecular limit is reached.
After noticing that the computational values of the force diminished in increasingly
larger chambers, the entire experimental setup was moved into a 3m diameter vacuum
77

Figure 6.1: Temperature gradient in the gas as a function of chamber
chamber (CHAFF). Experimental data for the larger chamber was taken in much the same
way, but due to the facility change the upper boundary of the pressure measurement was
drastically reduced from 6 Pa to1 Pa. Results for nitrogen in the large chamber are
presented in Fig. 6.3. While similar trends are observed when compared with the small
chamber data, it is obvious that the chamber itself has a dramatic eect. For the large
chamber, the peak of each device occurs at approximately the pressure of the peak in
the small chamber. There is also a noticeable decrease in the maximum force, as the
largest measured data in the small chamber was 1.28x10-6 N/K while in the large it was
7.53x10-7 N/K.
The reduction in force due to increasing chamber size is better shown in Fig. 6.4
where chamber comparisons for each geometric conguration are shown. For each device
the data behaves as predicted in the low pressure region, where it can be presumed that
vortex and collisional contributions are minimal. As the pressure increases the force
78

Figure 6.2: Force as a function of chamber size
becomes noticeably stronger in the smaller chamber. This makes physical sense when
compared with the limiting case of a chamber that is only slightly larger than the device;
the device will behave as if it were collisionless so long as the mean free path of a gas
molecule is longer than the distance between the device and the chamber wall.
Further clarication is provided in Fig. 6.5, where the ratios of curves t to the data
sets for each device are compared. It is clearly shown that the larger plates behave very
similarly. Not much can be concluded about the slight deviation between them in the
middle of the pressure range as this falls within the bounds of experimental uncertainty.
There is however a signicant deviation from the small plate, presumably caused by the
fact that the small plate is less aected by the shrinking chamber. This occurs because
the larger plates are proportionally closer to the chamber wall, and thus are more likely
to be aected by it.
79
0.00E+00
2.00E-07
4.00E-07
6.00E-07
8.00E-07
0 0.2 0.4 0.6 0.8 1
Pressure (Pa)
Force/ΔT, N/K
Circular Plate CHAFF
Large Plate CHAFF
Small Plate CHAFF

Figure 6.3: Comparison of device geometries in the 3.0m chamber (CHAFF)
80
0.00E+00
2.00E-07
4.00E-07
6.00E-07
8.00E-07
1.00E-06
1.20E-06
1.40E-06
0 0.2 0.4 0.6 0.8 1
Pressure (Pa)
Force/ΔT, N/K
Small Chamber
CHAFF
FM Prediction

(a) Circular device
0.00E+00
2.00E-07
4.00E-07
6.00E-07
8.00E-07
1.00E-06
1.20E-06
1.40E-06
0 0.2 0.4 0.6 0.8 1
Pressure (Pa)
Force/ΔT, N/K
Small Chamber
CHAFF
FM Prediction

(b) Large rectangular device
0.00E+00
1.00E-07
2.00E-07
3.00E-07
4.00E-07
5.00E-07
6.00E-07
7.00E-07
8.00E-07
0 0.2 0.4 0.6 0.8 1 1.2
Pressure (Pa)
Force/ΔT, N/K
Small Chamber
CHAFF
FM Prediction

(c) Small rectangular device
Figure 6.4: Comparison of device geometries in the 3.0m chamber (CHAFF)
81

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
Pressure (Pa)
FCHAFF / FSmallChamber
Small Plate
Large Plate
Circular Plate
Figure 6.5: Ratio of curve t data for a given device geometry
82
Chapter 7
Multiple Vanes: A Preliminary Study of Force
Optimization
It has been suggested[42] that an optimized vane structure may increase the performance
of a radiometer, and perhaps allow operation into much higher pressure regimes than
previously believed possible. To test this idea, a preliminary study has been conducted
to explore the eects of nearby radiometers on one another. To begin, it is worthwhile
to compare the
oweld from Figure 5.2 with those of other vane congurations. Four
separate computational results are shown in Figure 7.1, where they are presented in such
a way as to allow for easy comparison. All of these are 2-D half-space simulations of argon
gas, with the axis of symmetry lying along the line y=0, the initial pressure nominally at
0.5Pa, and the hot side of the vane facing left. The background of each plot displays the
density gradient, while the dark lines with arrows represent streamlines calculated from
the locally averaged velocities in a cell. The upper left plot has ve 10cm radiometer vanes
separated by a gap of 15cm between each vane, while the upper right plot shows a large
single vane with a height of 1.1m. The lower left one shows a three vane conguration
with separations of 40cm, and the lower right consists of two vanes with a 90cm gap.
83
Figure 7.1: ES-BGK streamlines and density eld in argon (0.5Pa) for multiple congu-
rations.
If the two vane conguration (lower right) is compared to the previous gure, only
small dierences are noticed. It is worth noting here that even though dierent gases
are used at dierent pressures, a comparison of these two gures is valid as both devices
are operating at approximately the same Knudsen number. Four distinct circulations
are formed in both cases, with the gas moving away from the centers of each plate and
returning near the edges. Also, the hot side circulations are noticeably stronger in both
cases, with the only visible dierence between them being the slight asymmetry of the

ow in the two vane case. When the two vane results are compared with those of the
three vane(lower left), a similar conclusion can be reached. Despite halving the distance
between successive vanes, the
oweld of each device appears to remain intact. Again,
84
the hot side vortex remains dominant and something very close to a symmetric
oweld
is created.
The ve vane conguration shows dierent results from the previous comparisons,
where the observed symmetry in the
oweld completely disappears. In this case it
appears that the
owelds have merged, where the previous
oweld structures seemed
to have very little impact on one another. Perhaps even more interesting than the merging
of the
owelds is the similarity that it seems to have with the large single vane, where
one can begin to see the beginning of the dominance of vortex on the hot side of the
upper most vane and the diminishing size of those closer to the centerline. Visualizing
the limiting case, one can see that this is precisely what would need to occur; as more
and more vane area is added, the impact on the
oweld from the vacant space would be
reduced until a device with innitely small spacing between vanes behaved exactly like
one that was solid.
Looking solely at the large single vane case (top right) it seems as if something in-
consistent is occurring. Though a large circulation still exists near the edge of the plate,
a counter propagating one has also formed near the centerline. As it is easy to imagine
that a large single device would operate in much the same way as a smaller single vane,
this seems slightly odd. However, the size of the large device is approximately an order
of magnitude bigger than the the 10cm vanes, so it will obviously reach transition sooner.
It is not exactly clear if the single large vane should have multiple circulations as shown,
or if it should have only two above the centerline (one hot, one cold) as in the case of
the one, two, and three vane simulations. Though it is possible that this is a condition
created by the symmetry plane and indicates some problem with the computation, it is
85
more likely that this is an example of how the proximity of the chamber wall aects the
behavior of the radiometer.
Viewing the temperature proles in Fig. 7.2, one can come to similar conclusions. It
is readily observed in the two vane case that there is minimal impact due to the presence
of a second radiometer at such a large distance. Even at half the separation the three
vane case only shows a minimal amount of interaction between the vanes, where it can
probably be assumed that this separation is somewhere around the minimum needed for
the vanes to begin interfering with each other's operation. For the ve vane case it is
obvious that the temperature proles are strongly interacting, which somewhat explains
the complicated
oweld observed above. Comparison of the ve vane case with the large
single vane further supports the theory that the addition of more vanes will trend toward
the limit of the large single one, as the temperature prole near the walls looks strikingly
similar and appears to be converging near the vanes.
One additional comparison of value is shown in Figure 7.3, where the pressure proles
of each case are shown. In each of these plots the local pressure is normalized by the
minimum background pressure of each case, and the results are presented as a percentage
representing the local overpressure. A dierent normalization pressure was applied to
each simulation as the heat
ux from the plate into the gas was dierent for each case,
with the single vane having the largest background pressure and the two vane one having
the lowest. For reference, the normalization pressures were 0.526, 0.541, 0.558, and
0.573Pa for the two, three, ve, and single vane simulations, respectively. It is clearly
seen, especially in the single vane case, that the largest overpressure occurs very near the
edge of the vanes on the hot side. From the single vane plot is is also readily observed
86
Figure 7.2: ES-BGK temperature eld in argon (0.5Pa) for multiple congurations.
that a nearly uniform pressure prole is created along the vast majority of the vane
surface, which diers somewhat from the historical predictions of Einstein. As previously
discussed, he stated that the only eective part of the radiometer is an area one mean
free path wide at the very edge of the vane. Once more, comparison of the lower two
plots suggest that there is little in
uence of one vane on another at separation distances
greater than 4 vane dimensions. As before, the results of the ve vane simulation again
trend toward that of the single large vane.
Though the general trends observed above show the convergence of the
oweld and
pressure proles with an increasing number of vanes toward that of the single vane, the
force production of the dierent cases vary dramatically. This is shown in Figure 7.4,
where the four previous cases are presented with the addition of a nine vane case at a
87
Figure 7.3: ES-BGK normalized pressure eld in argon (0.5Pa) for multiple congura-
tions.
nominal pressure of 0.5 Pa. The plot shows that the force production of an individual vane
behaves somewhat unexpectedly, where the eciency of the device is governed largely by
its proximity to another. In fact, from an eciency standpoint, a case can clearly be
made that an optimal vane separation exists somewhere in the range of 0.25 to 4 vane
lengths. This can be observed by comparing the normalized dierential pressure of all
ve congurations at the outermost vane location (0.50:05m). Here, the single vane is
shown to produce the lowest forces locally, mainly due to the completely solid nature of
the device.
Interestingly, the two and three vane congurations produce very similar local forces,
where the center and left sides of the vanes are more active than in the single case. This
88
Figure 7.4: ES-BGK normalized dierential pressure prole in argon (0.5Pa) for multiple
congurations.
provides further evidence that vane separations of more than ve vane lengths can be
mostly ignored. More notable perhaps is the distinct increase noticed in the ve vane
case, where the local force production of the outermost vane is noticeably higher than
that of the two, three, and single vane cases. This is unexpected for two reasons: rst, it
implies that the trend toward that of the single vane is not straightforward, and second,
that there is some mechanism previously unaccounted for that increases force production
independent of both area and perimeter. Finally, comparison with the nine vane results
demonstrates that as vane separation continues to decrease, an ultimate maximum is
achieved and local force production begins to decline.
This trend is also shown in Figure 7.5, where the force per unit length is shown
for a wide range of background pressures. As is expected, the free molecule results at
0.00625Pa show that nearly identical forces are produced by the vanes regardless of their
89
conguration or separation distance. As the pressure increases, the dierent eciencies
become readily apparent. Shown in Figure 7.4 as a function of distance, this plot presents
the force as a single value which has been integrated over the surface of the outer vane
and divided by its height (10cm in all cases but the single vane).
Figure 7.5: ES-BGK vane force for multiple pressures and congurations.
It should be noted here that despite the increased eciency of the smaller vanes when
compared to the large single vane, Figure 7.6 shows that the number of vanes is still
important to the total force produced. Here, the force produced by each vane is added
up and plotted as a function of background pressure. It is clearly shown that the single
vane (more vane area, less vane perimeter) outproduces both the two vane and three
vane simulations by a large margin. Notably, both the ve and nine vane congurations
outperform the single vane.
90
Figure 7.6: ES-BGK force per unit length of vane for multiple pressures and congura-
tions.
Perhaps the most important conclusion to be made from these results is that the
problem at hand has an extremely large scope. It has been shown here via comparison
of the single vane case with the others, that some combination of total vane area and
perimeter must responsible for force production. This agrees well with data shown in
the other chapters of this work, and is a non-trivial result as it shows that the perimeter
dependence of most theoretical work is not accurate when considering forces near the
maximum. It is worth noting the results these multi-vane
owelds provide, as they
conrm that there are several aspects of the radiometer which are important to its opera-
tion. First, comparison of the helium
oweld(Fig. 5.2) and the argon
owelds(Fig. 7.1)
suggest that dierent gases have similar behavior but at dierent pressures, as has been
shown in Chapter 4. Second, the necessary trade o between area and perimeter depen-
dence shown in Chapter 5 is a valid concern, especially because of the apparent usefulness
91
of the central part of the vane. Finally, the
oweld structure of the large single vane
suggests that the size of the chamber in relation to the device likely has a strong eect
on both the
oweld and the total force produced, as presented in Chapter 6.
92
Chapter 8
Conclusion
A study of the radiometric forces on heated plates has been conducted both experimen-
tally and computationally. The experiments were carried out at USC in two vacuum
chambers, up to a maximum pressure of 6 Pa for various gases, including: air, argon,
carbon dioxide, helium, nitrogen, and xenon. Multiple force measurements were taken
using an array of shapes and attachment mechanisms, where the surface temperatures
of both sides of the vanes were accurately measured for each data point. The compu-
tations were performed with both the DSMC and the ES-BGK methods for a 2-D gas

ow over a comparable range of pressures. In addition, axissymmetric calculations were
used to obtain extremely accurate results for a 3.0 m diameter chamber, with which the
experimental measurements were compared.
It is shown that the radiometric devices provide maximum force at a Knudsen number
approximating 0.1, that force production scales linearly with temperature dierential, and
that the force production of a radiometer vane is signicantly dependent on gas species.
Of the various gases tested, helium provides the largest peak force, entirely due to the
small collisional cross section of the molecule. A comparison of the force measurements for
93
all of the examined gases shows that the peak force production is inversely related to the
molecular cross section, where a larger cross section produces a smaller total force. This
occurs due to the higher pressure at which a device will transition, and the complicated
dependence of force production with background pressure.
Direct comparison of experiment and computation yield several conclusions; primarily,
a lack of experimental data on gas-surface accommodation and
ow three-dimensionality
yields up to a 40% dierence in the magnitude of the measured and computed forces.
It is possible, however, to use axissymmetric calculation and experimentation to negate
these dimensional problems, and to use the remaining discrepancy between the methods to
calculate the accommodation coecient of the carrier gas with the vane surface. Here, the
computed values of for argon and helium on aluminum are 0.79 and 0.50, respectively.
These fall within the wide experimental range of values put forth historically, and compare
favorably with modern measurements as well.
Qualitatively, the experimental data and computational results are also similar. Com-
parison of four solid geometric congurations has shown that the eect of the vane area
is signicant at pressures up to, and beyond, where the force is maximum. Specically,
it is shown via computation that the force created by the vane propagates inward from
the edge of the device over several mean free paths. This runs counter to most historical
treatment of the device, where it has been suggested that the eective area of a radiome-
ter is merely a small band near the edge of the device one free path wide. In fact, near
the maximum force a dierential pressure prole is shown to be nearly
at, such that
the higher pressures near the edges do not contribute signicantly more force than the
center of the vane. Further proof of the vane-area dependence of the force is provided
94
from the experimental results, where comparison of a rectangle and a circle of equal area
but unequal perimeter produce experimentally identical forces. Further comparison with
devices of smaller area and perimeter do suggest some perimeter dependence, but it is
again shown that the impact of vane area is by no means negligible.
Seemingly somewhat counter to the area dependence argument, a series of 2-D ES-
BGK computations in a small chamber suggest that several vanes in close proximity to
each other outperform a single large solid vane. Simulated congurations of two, three,
ve, and nine separate 10 cm vanes in a 0.5x1.6 m chamber show that the total force
production of a system of vanes varies widely; the two and three vane congurations
under perform the large single vane (1.1 m)while the ve and nine vane cases outperform
it by as much as a factor of 2.3. Interestingly, it is shown that this out performance
is not due entirely to an increase in device perimeter so much as to the proximity of
one vane to another. Indeed, comparison of the dierent congurations shows that force
production per vane has some maximum such that there is an optimal spacing between
vanes which lies somewhere between 4 vane dimensions and 0.25 vane dimensions. Though
computational cost precluded a more in-depth look at optimization, this is one area in
which further work is denitely warranted. Also notable is that the term 'optimization'
here can mean dierent things; plots of the total force and force eciency suggest that
there are dierent ways to optimize such a device. If eciency is the primary goal of the
device, the device would likely have vane separations in the range listed above. However,
if total force is the desired metric, the results seem to imply that tightly packed vanes
are the ideal way to achieve this. Finally, it seems logical to believe that some limit must
exist describing how small the gap between vanes can be before the separate vanes begin
95
behaving as if they are connected, but this distance has not been found in these limited
computations.
It must be noted here that it has also been demonstrated that the size of the chamber
in which the radiometer resides is of primary importance. Here, the size of the chamber is
inversely related to the generated force, such that small chambers produce larger measured
forces. This is shown experimentally as a ratio of forces for each device type in a pair of
vacuum chambers, as well as computationally for a single device in a domain of various
dimensions. With such a dependence, the chamber has had a measurable impact on
all simulated and experimentally obtained results. Though this dependence was found
to drop o exponentially such that chambers larger than 3.0m are believed to have a
negligible eect, there is no doubt that the results in smaller vessels are dramatically
aected by their proximity to a boundary. Under these circumstances it is important
to realize that the absolute applicability of the results presented in this thesis must be
couched in the knowledge that a chamber was used; the measured forces are valid only
under these specic conditions, with the caveat that the large chamber results are likely
the most applicable to a device operating outside of a laboratory in the upper atmosphere.
With regard to the multi-vane simulations presented previously, it must be noted that
the small size of the simulation domain is likely exaggerating the dierences between the
various cases, and that no experimental testing has yet been done to verify their validity.
While the results so far are encouraging, they are far from complete. Although many
improvements to historical measurement techniques have been incorporated in this work,
further improvement of accuracy is almost always desirable. Especially with regard to
96
measurement of the accommodation coecient, several things are in need of rethink-
ing. The rst of these is the temperature measurement of the vane surfaces, as the vast
majority of the experimental uncertainty came from temperature normalization of the
data. This could be accomplished in several ways, the most intuitive of which are accu-
rate and active temperature control of the vanes, along with more precise temperature
transducers such as platinum resistance thermometers (PRTs) or thermistors. Beyond
that, a repeatable method of surface preparation combined with some way of measuring
cleanliness would no doubt be of use. As mentioned above, there is obvious value to
an optimization study regarding maximum eciency and maximum force production of
a multi-vaned radiometer conguration, where signicant computer time will be needed
to solve a large computational matrix of vane congurations in much bigger domains.
Finally, experimental devices with novel vane structures should be investigated to add to
the body of measured data, and to verify the results of these computer simulations.
97
References
[1] A.A. Alexeenko, S.F. Gimelshein, E.P. Muntz, and A.D. Ketsdever. Kinetic modeling
of temperature-driven
ows in short microchannels. Int. J. Thermal Sciences, in
press.
[2] Darryl J. Alofs, Richard C. Flagan, and George S. Springer. Density distribution
measurements in rareed gases contained between parallel plates at high temperature
dierences. Phys. Fluids, 14(3):529{533, March 1971.
[3] K. Aoki, K. Kanba, and S. Takata. Numerical analysis of a supersonic rareed gas

ow past a
at plate. Phys. Fluids, 9(4), 1997.
[4] Gregory Benford and James Benford. An aero-spacecraft for the far upper atmo-
sphere supported by microwaves. Acta Atronautica, 56:529{535, 2005.
[5] Abraham Bennet. A new suspension of the magnetic needle invented for the discov-
ery of minute quantities of magnetic attraction. Phil. Trans. R. Soc. London, 82:81,
1792.
[6] P.L. Bhatnagar, E.P. Gross, and M. Krook. A model for collision processes in gases.
Phys. Rev., 94:511{525, 1954.
[7] G.A. Bird. Molecular Gas Dynamics. Clarendon Press, Oxford, 1976.
[8] G.A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows.
Clarendon Press, Oxford, 1994.
[9] E. Brueche and W. Littwin. Experimental contributions to the radiometer question.
Zeitschrift fur Physik, 52:318{335, 1928.
[10] William Crookes. On attraction and repulsion resulting from radiation. Proc. R.
Soc. London, 163:277, 1873.
[11] William Crookes. On attraction and repulsion resulting from radiation. Proc. R.
Soc. London, Sev. A, 23:373{378, 1875.
[12] C.W. Draper. The crookes radiometer revisited. J. Chem. Ed., 53(6):356, 1976.
[13] A. Einstein. Z. Physik, 27:1, 1924.
[14] Epstein. Z. Physik, 54:537, 1929.
98
[15] Augustin Fresnel. Annales de Chimie et de Physique, 29:57, 1825.
[16] M.A. Gallis. private communication. September, 2008.
[17] B. Gotsman and U. Durig. Experimental observation of attractive and repulsive
thermal forces. Applied Physics Letters, 87, 2005.
[18] Haade. http://en.wikipedia.org/wiki/File:Crookes radiometer.jpg, March 9 2003.
[19] Stewart Harris. An Introduction to the Theory of the Boltzmann Equation. Dover
Publications, Inc., New York, 2004.
[20] Hettner and Czerny. Z Physik, 27:12, 1924.
[21] L.H. Holway. Thermal accommodation coecient and adsorption of gases. 4th Int.
Rareed Gas Dynamics Proceedings, 1:193{215, 1964.
[22] F.C. Hurlbut. J. Appl. Phys., 28:844, 1957.
[23] M.S. Ivanov, G.N. Markelov, and S.F. Gimelshein. Statistical simulation of reactive
rareed
ows: Numerical approach and applications. AIAA Paper, 98:2669, June
1998.
[24] A.J. Jamison, A.D. Ketsdever, and E.P. Muntz. Gas dynamic calibration of a nano-
newton thrust stand. Review of Scientic Instruments, 73:3629{3637, 2002.
[25] E.H. Kennard. The Kinetic Theory of Gases. McGraw-Hill, New York, 1938.
[26] M Knudsen. Ann. Physik, 6:129, 1930.
[27] M.N. Kogan. Rareed Gas Dynamics. Plenum Press, New York, 1969.
[28] Leonard B. Loeb. Kinetic Theory of Gases. McGraw-Hill, New York, 1927.
[29] Leonard B. Loeb. Kinetic Theory of Gases(Third Ed.). Dover Publications, New
York, 1961.
[30] H.E. Marsh. Further experiments on the theory of the vane radiometer. J. Opt. Soc.
Amer., 12:135{148, 1926.
[31] J.C. Maxwell. A treatise on electricity and magnetism, volume 2. Clarendon Press,
Oxford, 1873.
[32] J.C. Maxwell. On stresses in rareed gases arising from inequalities of temperature.
Phil. Trans. R. Soc., 170:231{256, 1879.
[33] L. Mieussens. Discrete-velocity models and numerical schemes for the boltzmann-
bgk equation in plane and axisymmetric geometries. J. Computational Physics,
162(2):429{466, 2000.
[34] Luc Mieussens. Convergence of a discrete-velocity model for the boltzmann-bgk
equation. Computers and Mathematics with Applications, 41:83{96, 2001.
99
[35] Masahiro Ota, Tomohiko Nakao, and Moriyoshi Sakamoto. Numerical simulation of
molecular motion around laser microengine blades. Mathematics and Computers in
Simulation, 55:223{230, 2001.
[36] A. Passian, R.J Warmack, and et al. Observation of knudsen eects with microcan-
tilevers. Ultramicroscopy, 97:401, 2003.
[37] A. Passian, R.J. Warmack, T.L. Ferrell, and T. Thundat. Thermal transpiration at
the microscale: a crookes cantilever. Phys. Rev. Letters, 90(12):124503, 2003.
[38] A. Passian, A. Wig, F Meriaudeau, T.L. Ferrell, and T. Thundat. Knudsen forces
on microcantilevers. J. Appl. Phys., 92(10):6326, 2002.
[39] O. Reynolds. On the forces caused by the communication of heat between a surface
and a gas; and on a new photometer. Phil. Trans. R. Soc., 166:725, 1876.
[40] O. Reynolds. Of certain dimensional properties of matter in a gaseous state. Phil.
Trans. R. Soc., 170:727{845, 1879.
[41] S.C. Saxena and R.K. Joshi. Thermal Accommodation Coecient and Adsorption
of Gases. McGraw-Hill, New York, 1924.
[42] M. Scandurra. A novel thermoelectrically driven engine for air and ground vehicles.
D.O.E. Presentation, San Diego, 2004.
[43] M. Scandurra, F. Iacopetti, and P. Colona. Gas kinetic forces on thin plates in the
presence of thermal gradients. Phys. Rev. E, 75:1{5, 2007.
[44] Arthur Schuster. On the nature of the force producing the motion of a body exposed
to rays of heat and light. Proc. R. Soc. London, 24:391{392, 1875.
[45] N.P. Selden and A.D. Ketsdever. Comparison of force balance calibration techniques
for the nano-newton range. Review of Scientic Instruments, 74:5249{5254, 2003.
[46] Th. Sexl. Z Physik, 52:249, 1929.
[47] Y. Sone. Eect of sudden change of wall temperature in rareed gas. J. Phys. Soc.
Japan, 20(2):222, 1965.
[48] G.S. Springer. Heat transfer in rareed gases. Advances in Heat Transfer, ed. by
Irvine and Hartnett, Academic Press, New York:163{218, 1971.
[49] Robert E. Stickney. Momentum transfer between gas molecules and metallic surfaces
in free molecule
ow. Phys. Fluids, 5(12):1617{1624, December 1962.
[50] G.J. Stoney and R.J. Moss. On crookes' force. Proc. R. Soc. London, 25:553, 1877.
[51] W.P. Teagan and George S. Springer. Rev. Sci. Instr., 38:335, 1967.
[52] W.P. Teagan and George S. Springer. Phys. Fluids, 11:497, 1968.
100
[53] W.M. Trott, D.J. Rader, J.N. Castaneda, J.R. Torczynski, and M.A. Gallis. Mea-
surement of gas-surface accommodation. 26th Int. Rareed Gas Dynamics Proceed-
ings, 2008.
[54] Harold Y. Wachman. Thermal accommodation coecient: The contributions of
lloyd brewster thomas. RGD, 185:461, 1992.
[55] D.C. Wadsworth, N.E. Gimelshein, S.F. Gimelshein, and I.J. Wysong. Assesment of
translational anisotropy in rareed
ows using kinetic approaches. 26th Int. Rareed
Gas Dynamics Proceedings, 2008.
[56] Dean C. Wadsworth and E. Phillip Muntz. A computational study of radiomet-
ric phenomena for powering microactuators with unlimited displacements and large
available forces. J. MEMS, 5:59{65, March 1996.
[57] Gilbert D. West. A modied theory of the crookes radiometer. Proc. Phys. Soc.
London, 32:222{231, 1919.
[58] Wilhelm H. Westphal. Mesungen am radiometer. Zeitschrift fur Physic, 1:92{100,
1920.
[59] A.E. Woodru. William crookes and the radiometer. Isis, 57(2):188, 1966.
[60] J.Y. Yang and J.C. Huang. Rareed
ow computations using nonlinear model boltz-
man equations. J. Computational Physics, 120:323{339, 1995.
[61] Zedh. http://en.wikipedia.org/wiki/Linear variable dierential transformer, May
13 2007.
101

Abstract (if available)

Abstract A study of the radiometric forces on heated plates has been conducted both experimentally and computationally. The experiments were carried out at USC in two vacuum chambers up to a maximum pressure of 6 Pa for various gases. The computations were performed with both the DSMC and ES-BGK methods for a 2-D gas over a comparable range of pressures. It is shown that the radiometric devices provide maximum force at a Knudsen number approximating 0.1. Of the various gases tested, helium provides the largest peak force. Qualitatively, the experimental data and computational results are similar. A lack of experimental data on gas-surface accommodation and three-dimensionality yields up to a 40% difference in the magnitude of the measured and computed forces, but it is shown that this discrepancy can be used to predict accommodation values. Comparison of four geometric configurations has shown that the effect of the area is signficant at pressures up to where the force is maximum. It is also demonstrated that the size of the chamber in which the radiometer resides is of primary importance, where the chamber dimensions are inversely related to the generated force. Finally, simulation of multi-vane configurations have shown that the optimal spacing of vanes can be tailored for specfic uses

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

An approach to experimentally based modeling and simulation of human motion

PDF

Inference of computational models of tendon networks via sparse experimentation

PDF

Numerical study of flow characteristics of controlled vortex induced vibrations in cylinders

PDF

Mesoscale SOFC-based power generator system: modeling and experiments

PDF

Effect of surfactants on the growth and departure of bubbles from solid surfaces

PDF

Experimental and kinetic modeling studies of flames of H₂, CO, and C₁-C₄ hydrocarbons

PDF

Application of optical forces in microphotonic systems

PDF

Numerical and experimental investigations of dust-plasma-asteroid interactions

PDF

Development and applications of a body-force propulsor model for high-fidelity CFD

PDF

An approach to dynamic modeling of percussive mechanisms

PDF

High temperature latent heat thermal energy storage to augment solar thermal propulsion for microsatellites

PDF

Efficiency droop in indium gallium nitride light emitters: an introduction to photon quenching processes

PDF

Modeling and simulation of complex recovery processes

PDF

Computational fluid dynamic analysis of highway bridge superstructures exposed to hurricane waves

PDF

A process-based molecular model of nano-porous silicon carbide membranes

PDF

Flame ignition studies of conventional and alternative jet fuels and surrogate components

PDF

Lab-scale and field-scale study of siloxane contaminants removal from landfill gas

PDF

Experimental studies of high pressure combustion using spherically expanding flames

PDF

Resonant light-matter interactions in nanophotonic structures: for manipulating optical forces and thermal emission

PDF

Computational investigation of glutamatergic synaptic dynamics: role of ionotropic receptor distribution and astrocytic modulation of neuronal spike timing

Asset Metadata

Creator Selden, Nathaniel P. (author)

Core Title Experimental study of radiometric forces with comparison to computational results

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Publication Date 04/22/2009

Defense Date 01/28/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag accommodation,Einstein,gas surface interaction,Knudsen,low density gas,microfluidics,OAI-PMH Harvest,radiometer,radiometric forces,rarefied flow,thermal gradients,thermal transpiration

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Muntz, E. Phil (committee chair), Domaradzki, Julian A. (committee member), Gundersen, Martin A. (committee member), Shiflett, Geoffrey R. (committee member), Wang, Hai (committee member)

Creator Email nate@qvessel.com,selden@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2107

Unique identifier UC1482586

Identifier etd-Selden-2649 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-226080 (legacy record id),usctheses-m2107 (legacy record id)

Legacy Identifier etd-Selden-2649.pdf

Dmrecord 226080

Document Type Dissertation

Rights Selden, Nathaniel P.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu