CRYONICS
VOLUME 6(9) SEPTEMBER, 1985 ISSUE # 62
Editorial Matters..............................................page 1
Renewing Your Subscription.....................................page 1
Liability Insurance Debacle....................................page 2
Cryonics Coordinators -- Another Try...........................page 4
A Record-Breaking Month........................................page 5
ALCOR Education and Promotion..................................page 6
ALCOR Florida..................................................page 7
Moving Into The Vault..........................................page 8
Heat Flow In The Cryonic Suspension Of Humans..................page 9
Letter to The Editors..........................................page 34
The Scanning Tunneling Microscope..............................page 35
ALCOR Meeting Schedule.........................................page 38
CRYONICS is the newsletter of the ALCOR Life Extension Foundation, Inc.
Mike Darwin (Federowicz) and Hugh Hixon, Editors. Published monthly.
Individual subscriptions: $15.00 per year in the U.S., Canada, and
Mexico.; $30.00 per year all others. Group rates available upon request.
Please address all editorial correspondence to ALCOR, 4030 N. Palm St.,
#304, Fullerton, CA 92635 or phone (714) 738-5569. The price of back
issues is $2.00 each in the U.S., Canada, and Mexico, and $2.50 for all
others.
Contents copyright 1985 by ALCOR Life Extension Foundation, Inc., except
where otherwise noted. All rights reserved.
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EDITORIAL MATTERS
We are pleased this month to publish Art Quaife's paper Heat Flow in
the Cryonic Suspension of Humans. We have been after Art, who is the
President of Trans Time, to allow us to run this work, as we feel that it
is a valuable contribution to the literature of cryonics. To some extent,
our feelings about being able to publish it are mixed, since it means that
Art was unsuccessful in attempting to publish it in the regular scientific
literature. We feel that this is unfortunate to science as a whole, since
his work is likely to be quite useful in organ and limb preservation, and
we are not aware of any workers in those fields who are subscribing to
CRYONICS. We realize that many of our readers may not share our
enthusiasm, as they attempt to make sense of what is a rather abstruse
piece of mathematics. Notwithstanding, it is important. The cylinders and
spheres Art talks about approximate your arms, legs, body, and head, and
that's important to all of us.
We are also happy to say that the 1984 Index to CRYONICS, compiled by
Steve Bridge, is in the center of this issue. It has been sitting in our
"IN" basket for nearly three months, an embarrassingly long time.
RENEWING YOUR SUBSCRIPTION
About a month before your subscription to CRYONICS is due to lapse you
will receive a Post Card letting you know that it's renewal time. In the
past, we have sent out only one renewal notice. This has caused a problem
for a few people when the US Mail loses their renewal notice. We use post
cards and send out only one renewal notice primarily because it is
expensive to send out multiple notices (or even one notice, for that
matter) in first class envelopes. In fact, each such letter costs about
$1.00 (in postage, paper products and time) to produce and mail! Since
subscriptions to CRYONICS are only $15.00 a year, and we are already losing
money on publication, we've had no choice but to keep "incidentals" like
billing to a minimum.
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For the same reason we cannot "set back" subscriptions if you fail to
renew on time (such as by starting a subscription two or three issues
back). Our postage costs are about 6 cents per issue if we bulk mail and
over 40 cents per issue if we mail First Class! The Postal service is very
picky about bulk mailings -- all items must be identical. . . and they DO
check! It is simply not economically possible for us to mail back issues
out as part of a normal subscription. When we were doing this several
years ago it became a significant financial and administrative burden.
So, what can be done about people who miss their subscription renewal?
Well, we have decided to send out two notices for renewal, even if this
does mean some added cost. But, there's also something you can do. If you
don't get your magazine and you're wondering about your subscription
status, an easy way to find out is to look on any recent back issue of
CRYONICS which you have and check the upper right hand corner of your
mailing label. That number, which the computer put on, tells you when your
subscription expires(ed).
If you have missed some issues and you want to purchase them, they are
usually (we are now permanently out of stock on some back issues) available
and all you need to do to get one is send us $2.00 for each back issue you
want. Also, when you see that renewal Post Card in the mail, remember that
you have only one month to renew or you'll start missing issues!
LIABILITY INSURANCE DEBACLE
QUESTION: What do a gynecologist in Fullerton, a Jaycee's organization
in Michigan City, Indiana, a chemical company in Des Moines, Iowa, a day-
care center in Anaheim, the town of Tehema, California and the ALCOR Life
Extension Foundation have in common?
ANSWER: An insurance crisis. About 6-months ago Cryovita
Laboratories, where ALCOR shares quarters, was faced with what amounted to
a potentially disastrous crisis. In order to occupy the industrial bay in
which we are located and in order to do business in the City of Fullerton,
we are required to maintain a liability insurance policy to cover possible
injuries or accidents that might occur on our premises. Not only do we
have volunteers and guests come through our facility, we also have delivery
people, consultants, and occasionally contractors work at or visit our
premises. Every business is required to provide insurance to cover the
contingency of accident or injury. The ugly fact of the matter is that
even if someone breaks into our building and trips and falls over an
extension cord -- we're probably liable to some degree!
In order to keep our lease and our business license we have to have
that coverage. Period. In the past liability coverage has cost us about
$300.00 per year. Not a trivial sum for an organization as small as ours.
You can imagine our surprise when we were told that 1) it was very unlikely
that we could get any coverage at all and 2) if we did get coverage it
would probably cost upwards of $2,000! (Yes, that right -- two thousand,
not two hundred!)
Our insurance agent turned out to be wrong on both counts. We did
finally
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find coverage and it didn't cost upwards of $2,000 -- it cost upwards of
$3,000! Our agent phoned a few days ago to inform me that it was very
unlikely we will be able to get coverage at all next year. At any price.
When I asked him what we were going to do, he asked if we had half a
million dollars to post a bond or self-insure! Helpful advice!
The only comfort in all this is that we are not alone. Liquor stores,
charity picnics, even the Southern California Rapid Transit District have
been unable to get coverage. All across the United States many hundreds of
businesses are closing or, where the law permits (and even where it
doesn't) are "going bare" -- in other words going without liability
insurance.
Like ALCOR and Cryovita these businesses are "high risk." If a
suspension patient's family sues us, if an animal rights terrorist throws a
brick, bottle, or bullet through our window and someone is injured, the
insurance company may find itself in court. Leave aside the fact that
liability insurance for personal injury and property damage has nothing to
do with liability for suspension services. All the insurance companies
know is that they may find themselves in court -- and as anyone who has
been in court can attest, even if you win, you lose.
Who's to blame for this?: a suit-crazy segment of the American people
and the nation's trial lawyers. Litigation has become the easy path to
revenge and "riches." In assigning blame we come down hard and heavy on
irresponsible trial lawyers who litigate at the drop of a hat for damages
that are of Alice in Wonderland proportions. In our book, such trial
lawyers and the green slime of decomposition go hand in hand. Cryonics has
a long and unfortunate history of litigation. In large measure this has
been due to the presence of lawyers whose advice has hamstrung, paralyzed,
and even led cryonics organizations into situations certain to produce
contention, strife, and ultimately, litigation.
Where does all of this
leave ALCOR? Frankly, we don't
know. We are in the process of
trying to prepare an alternate
facility (in an unincorporated
area) which will be owned by
cryonicists. Failing this, we
may very well find ourselves out
on the street and out of busi-
ness next year when our policy
comes up for renewal. That is,
if we don't get cancelled mid-
year! According to the Califor-
nia Department of Insurance,
many carriers have started can-
celling policies and returning
the unused portion of the prem-
ium. A pleasant thought. Oh
well, nobody said living forever
was going to be easy. They said
it was going to be impossible.
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ALCOR COORDINATOR PROGRAM: ANOTHER TRY
In cryonics the distance between interest and participation is often
vast. Many times we receive requests for information which constitute
"good leads" but we lose the person because there's no opportunity for
follow-up and no way that individual can get involved on a local, grass-
roots level. Sometimes the situation is an urgent one and the caller needs
information immediately -- particularly access to printed materials and
personal contact from someone who is thoroughly knowledgeable about
cryonics, and that someone just isn't there.
A little over a year ago we attempted to solve this problem by
establishing a Cryonics Coordinator program. Sadly, we received little
interest, and the idea was shelved. This is unfortunate because a
Coordinator Program is a good idea, even a potentially life saving idea.
It is particularly frustrating to us to see people who are living in the
same city as subscribers or members -- and they don't even know about each
other. Obviously there's a lot of untapped talent and resources which
could be put to use on a local level if only someone was willing to
exercise the responsibility and leadership of "putting it all together."
We know that there are ALCOR Suspension Members scattered across the
U.S. who have the talent and ability to handle this job. We are frustrated
and disappointed that few of them have been willing to step forward and
take some responsibility, and would like to remind all our members that
ALCOR is a voluntary self-help organization. The share of ALCOR's burden
that each of us carries varies, and we accept this, within reason. It is
good, however, to know that you are not the only one out there with your
shoulder to the wheel, making sure this ALCOR of ours gets to its desired
destination. And it is not good to have the feeling that one is being
taken advantage of in this respect.
We would like to see a program of ALCOR Coordinators. Coordinators
would initially act as a local resource to refer information requests to
and, when the level of local interest seemed to justify it, to try and set
up meetings and get an informal or formal local group going. Literature
and leads would be supplied by ALCOR, as well as advice on handling the
media and/or pursuing local publicity -- should the Coordinator choose to
do so.
Groups formed in this way could do much,
even in the early days to improve their chances.
Just the presence of vocal, well informed people
at a location remote from ALCOR in California or
Florida could be a tremendous advantage if an
emergency situation were to develop. If a group
of even two or three forms, it will probably be
able to very quickly afford to maintain a local
heart-lung resuscitator (HLR) and emergency drug
box--greatly speeding response time should the
need arise. We estimate that a local resusc-
itation/stabilization capability, including HLR,
medications and other basic equipment could be
put in place for under $1,000! This should be
easily in reach of even a very small local
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group. And keep in mind that hospitals typically do not have HLRs or many
of the medications used in a suspension, and may be unwilling or legally
unable to administer them even if they do. In that situation, which is the
all too common one, the member will be faced with a wait of many hours
until we can arrive on the scene. A wait that would have been unnecessary
if help and equipment had been available on a local level.
So, we are going to give the Coordinator program another try. We want
to hear from you ALCOR members out
there and we want to see you meet us
half way. We are willing and ready
to go all-out for you, but we can't
do it alone.
The requirements for being an
ALCOR Coordinator are simple. You
must be an ALCOR Suspension Member,
you must be willing to take
referrals and talk to people who are
interested in cryonics, and you must
spend some time talking with us, so
that we can pass on our experiences
and evaluate your ability to do the
work. We ask that we be allowed to
list your name, city and address (a
P.O. Box is fine) periodically in
CRYONICS. This is important, since
we know from experience that many
people will contact someone locally
much more readily than they will
send off to California for informat-
ion. Don't let this opportunity
slip by. Let us hear from you! If
you would like to be an ALCOR Coord-
inator call us or write us today.
A RECORD-BREAKING MONTH!
As we've mentioned several times in the recent past, ALCOR has been
signing up new suspension members at the rate of about 3 per month for the
past 6 months or so. August was an incredible exception to that average:
12 new suspension members were approved by the Board of Directors at its
August 6th meeting! As far as we know this is an all time record. Perhaps
more exciting is the fact that the overwhelming majority of these new
members are people who are "new" to cryonics as well -- they only recently
heard of us or they have never been signed up before.
We're realistic enough to know that we probably won't keep signing up
people at this rate steadily. There will be ups and downs, and it's
important not to plan the future on the basis of something as uncertain as
adding new members. On the other hand, one message is coming through loud
and clear: we're growing again and the new blood feels good!
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To all you new ALCOR members, Welcome Aboard, the adventure has just
begun!
ALCOR EDUCATION AND PROMOTION
Our growth in member-
ship tells us that we're
on the right track, but we
know that if cryonics is
to succeed we can't sit
back and rest on our acco-
mplishments. In any abso-
lute sense what we've ach-
ieved so far is but the
tiniest fraction of what
needs to be done. We need
to be signing up not mere-
ly 10 members a months, but 10,000. That day
is a long way off, and it won't get any closer
without a lot of hard work. To this end ALCOR has
been investing a fair amount of time and energy in
promotion -- and it seems to be paying off.
We are doing several radio shows a week, and
we are in the final stages of preparing a variety
of ads and media information packages for further
promotion. An ad featuring ALCOR Suspension Member
Saul Kent appears elsewhere in this issue. It is
as an example from a series of such profile ads
which we will be running. The second in this
series features ALCOR Suspension Member Dick Clair,
three time Emmy Award winner and creator of such
popular television series as FACTS OF LIFE, IT'S A
LIVING, FLO, and MAMA'S FAMILY. These ads will be
run in a variety of print media and will be mailed
out as flyers to appropriate mailing lists.
We have put together a preliminary ALCOR Press
Kit to use in media promotion of our activities and
we are working on several major media projects at
the current time. With luck and a lot of hard work
you should be hearing and seeing more of us in the
media over the next year or so.
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ALCOR FLORIDA
Mike Darwin spent 8 days in Florida, July 18 to July 26, conducting
another training session and touching bases with members there. This was a
particularly productive trip from both a working and promotional
standpoint. Mike was on several local radio shows, including the 3-hour
Allen Burke Show. Burke is a nationally known talk show host (who at one
time challenged Johnny Carson for supremacy of the evening airwaves) known
for his acidic treatment of guests. Mike managed not only to hold his own,
but to actually control the interview. As a result, South Florida got a
three hour long introduction to cryonics and ALCOR.
On Friday night, the day prior to the first training session, Bill
Faloon and Tina Lee hosted a get-together in their home. This was a
pleasant evening with some people driving in from as far away as West Palm
Beach and Cocoa Beach. It was a great opportunity for people to get to
know each other in a "nontechnical" setting. And, as you might guess, the
philosophical discussions went on into the wee hours of the morning.
Fortunately, the training session was not scheduled to start until noon the
next day, so team members had a little extra time to recover.
The two-day training session was used to review medications, I.V., and
respiratory support techniques as well as serve as an introduction to
perfusion operations. Mike Darwin took the opportunity to "solo" on
initiating femoral bypass with a bubble oxygenator and the Florida crew got
an introduction to sterile technique and bypass procedure. We're happy to
report that the training animal, a beautiful shepherd/huskie mix with a
lovely disposition, weathered three hours of bypass (and a lot of
intubation practice) and cooling to 22ųC beautifully. The dog was adopted
by Bill Faloon and Tina Lee and has been christened "Sandy." We understand
she is now running the Faloon/Lee household and dining exclusively on
cooked steak!
The Florida team continues to make progress, although several people
were unable to make this training session. This is cause for some concern,
since training sessions with personnel from California present are less
frequent than is desirable to maintain a high level of skill. Under these
circumstances, every session counts, and its important not to miss even one
training session.
There was also a lot of discussion about the future of cryonics in
Florida. Long-term strategies for facilities acquisition, marketing, and
growth were laid down, and contacts were established which will help
substantially in expanding operations in the future. All in all it was a
great trip.
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MOVING INTO THE VAULT: A BUSY WEEKEND
The day following Mike Darwin's
return from Florida, work began in ** PHOTO SPACE **
earnest to gear up for moving the ** CAPTION --
patients into the Cephalarium Vault.
A few recent contributions pushed "No sweat. The top of the
Frosty Thermometer over the top, and vault is lifted off with
at long last the time and money were Jerry Leaf aboard. Hugh
there to increase patient security to Hixon at the controls of
an all- time high. 6,000 lb. forklift/crane."
On Friday, July 26th, the five **
ALCOR neuropatients were transferred
out of the A-2542 storage dewar into
two back-up dewars. Ten days prior to
the transfer Hugh Hixon had "fired up"
the backup dewars and carried out
boil-off evaluations to make sure they
were in good working order and safe to
use. The A-2542 was then drained of
liquid nitrogen, warmed up, and
cleaned out. It's simply amazing how
much water vapor and particulate
debris accumulates in a working dewar
after almost four years of continuous
operation! Another reason for a warm-
up and clean-out of the A-2542 is to
get rid of the liquid oxygen (LOX)
which tends to accumulate in the
liquid nitrogen with time.
Once the A-2542 was cleaned out,
it was bolted to a shock absorbing
platform and readied for placement in
the vault. On Sunday an ALCOR crew
consisting of Hugh Hixon (the team
leader), Jerry Leaf, Scott Greene,
Brenda Peters, Mike Darwin, and a
massive, rented forklift with a "slip-on" crane plucked off the top of the
vault (which weighs a mere 1,500 pounds!) and lowered the dewar/support
platform into the vault. Brenda served as photographer with some help from
Scott Greene (who divided his time between snapping photos and assisting
with the moving operations). Brenda's pictures turned out great! A few of
them appear with this article and the technical article on the vault which
we will publish later. These pictures are somewhat unusual in that Hugh
Hixon appears in them. Usually he is both a worker and the photographer,
and hence invisible to the camera.
Jerry Leaf did a superb job managing the forklift and Mike Darwin
performed
(Article continued on page 31.)
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HEAT FLOW IN THE CRYONIC SUSPENSION OF HUMANS
SURVEY OF THE GENERAL THEORY
Arthur Quaife, President
Trans Time, Inc.
1507 63rd Street
Emeryville, California 94608
ABSTRACT
Procedures used in the successful freezing and thawing of diverse human
cells and tissues are known to be quite sensitive to the cooling and
thawing rates employed. Thus it is important to control the temperature
descent during cryonic suspension of the whole human body. The paper
surveys the general theory of macroscopic heat flow as it occurs during the
cryonic suspension of human patients. The basic equations that govern such
heat flow are presented, then converted to dimensionless terms, and their
solutions given in geometries that approximate the human torso, head, and
other regions of the body. The solutions are more widely applicable to the
freezing of tissues and organs.
1. INTRODUCTION
Cryonic suspension is the freezing procedure by which human patients
are preserved, after pronouncement of legal "death," in hopes of eventual
restoration to life and health. The procedure attempts to preserve the
basic information structures that determine the individual's identity.
These include the memories and personality as encoded in the macromolecules
and neuronal weave of the brain, and the genetic information stored in DNA.
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HEAT FLOW IN CRYONIC SUSPENSION
The author has previously formulated a mathematical model of the heat
flow and the diffusion of cryoprotectant that occurs during the first phase
of this procedure, in which chilled blood substitutes and cryoprotective
solutions are perfused through the vascular system [9]. The present paper
treats the general theory of heat flow, particularly at sub-zero
temperatures after perfusion has ceased and the body has solidified.
The author has written a computer program that calculates most of the
solutions given below, and in subsequent articles intends to present tables
and graphs comparing theoretical projections with experimental data. Other
problems for subsequent analysis include change of phase, and the thermal
stresses from temperature gradients within the frozen tissue.
2. NOMENCLATURE
We first describe the typographical conventions to be used. Beginning
in Section 4, all dimensioned variables, constants, fields, and operators
will be converted to dimensionless counterparts. Therefore we will use
ordinary roman type font for dimensioned real quantities (x), with roman
boldface to represent dimensioned vectors (*). Dimensionless real
quantities will be represented in italics (**), while dimensionless vectors
will be in boldface italics (***). Dimensioned constants in greek along
with the gradient operator will be unemphasized, while their dimensionless
counterparts will be in boldface (****).
The variables and fields used to describe heat flow within a solid
region R (such as the frozen human body) bounded by a surface S (such as
the skin) are:
Variable Description Unit
r' position vector m
t time s
T(r',t) temperature scalar field K
2
q'(r',t) heat flux vector field J/(m s)
3
g(r',t) rate of heat generation J/(m s)
where we use the standard abbreviations for SI units: m = meters, kg =
kilograms, s = seconds, K = degrees Kelvin, J = Joules [ = kg mż/sż].
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TYPIST'S NOTES: BECAUSE OF ASCII'S LIMITATIONS, BOLDFACE, ITALICS, AND
SOME GREEK LETTERS CANNOT BE USED HERE AS THE AUTHOR ORIGINALLY INTENDED.
MODIFIED CONVENTIONS WILL BE AS FOLLOWS:
(*) "ROMAN BOLDFACE" WILL BE REPRESENTED BY AN APOSTROPHE FOLLOWING THE
VARIABLE (r').
(**) "ITALICS" WILL BE REPRESENTED BY A DOUBLE APOSTROPHE FOLLOWING THE
VARIABLE (x'').
(***) "BOLDFACE ITALICS" WILL BE REPRESENTED BY A SINGLE QUOTATION MARK
FOLLOWING THE VARIABLE (r").
(****) GREEK CHARACTERS, ORIGINALLY UNEMPHASIZED OR BOLDFACE, WILL BE
REPRESENTED BY THEIR ASCII EQUIVALENTS WHERE POSSIBLE. WHERE NOT POSSIBLE,
THEY WILL BE SPELLED OUT BETWEEN BRACKETS ({rho}), AND FOLLOWED WITH AN
APOSTROPHE WHERE BOLDFACE ({rho}').
THE TYPIST OFFERS HIS APOLOGIES TO THE ARTICLE'S AUTHOR IF THIS
NOMENCLATURE DETRACTS FROM THE INTENDED CONCEPTS. REPRINTS OF THE ORIGINAL
ARTICLE CAN BE OBTAINED FROM ALCOR LIFE EXTENSION FOUNDATION.
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HEAT FLOW IN CRYONIC SUSPENSION
1
The constants needed to describe heat flow within a solid are:
Constant Description Unit
3
V volume of (finite) region R m
2
A area of (finite) surface S m
2
k thermal conductivity J m/(m s K)
3
{rho} density kg/m
c specific heat at constant pressure J/(kg K)
2
h heat convection at the boundary J/(m s K)
2
ą thermal diffusivity m /s
k
= ÄÄÄÄÄÄ
{rho}c
2
The total heat flow F (r',t) [J/m ] per unit area up to time t,in the
n
direction of unit vector n", is given by:
ōt
F (r',t) = ³ n"śq'(r',s) ds (2.1)
n õ0
Equation (2.1) may be integrate over a surface S having normal vector n" to
obtain the total heat flow across the surface.
The average temperatures within the region R and on the surface S are
given respectively by:
1 ō ō ō
= Ä ³ ³ ³ T(r',t) dV (2.2)
R V õ õ õR
1 ō ō
= Ä ³ ³ T(r',t) dA (2.3)
S A õ õS
The heat energy Q [J] contained within the region R, with respect to a
reference temperature T is determined from the above constants by:
ģ
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(1) k, c, and to a small degree {rho}, generally depend upon temperature.
By consulting tables showing their temperature variation within the
specific medium, we can determine the temperature range within the
"constant" approximation is appropriate.
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HEAT FLOW IN CRYONIC SUSPENSION
ō ō ō ōT(r't)
Q(t) = ³ ³ ³ ³ {rho}c(T) dT dV (2.4)
õ õ õR õT
ģ
Here we have explicitly shown the temperature dependence of c upon T, but
except for the solution presented in Section 11, we will otherwise assume c
to be a constant independent of temperature. In this case,
Q(t) = {rho}cV( - T )
R ģ
We also use the differential operators:
Operator Description Unit
gradient 1/m
{delta} {delta} {delta}
= i" ÄÄÄÄÄÄÄÄ + j" ÄÄÄÄÄÄÄÄ + k" ÄÄÄÄÄÄÄÄ
{delta}x {delta}y {delta}z
ż Laplacian 1/mż
{delta}ż {delta}ż {delta}ż
= i" ÄÄÄÄÄÄÄÄÄ + j" ÄÄÄÄÄÄÄÄÄ + k" ÄÄÄÄÄÄÄÄÄ
{delta}żx {delta}ży {delta}żz
when these are expressed in rectangular coordinates.
** TYPIST'S NOTE: APOLOGIES TO PURISTS, WHO WILL PROTEST THE USE OF THE
"" IN PLACE OF THE STANDARD INVERTED TRIANGLE OPERATOR SYMBOL. **
3. EQUATIONS GOVERNING HEAT CONDUCTION
For a homogeneous isotropic solid (a solid that has the same thermal
properties at every position, and in which the thermal conductivity is the
same is every direction), Fourier's law of heat flow is:
q'(r',t) = -kT(r',t) (3.1)
This law simply states that heat flows from position of high temperature to
positions of low temperature, and flows in the direction in which the
temperature decreases most rapidly, in proportion to the rate of decrease.
In addition we have the continuity equation:
{delta}T(r',t)
śq'(r',t) - g(r',t) + {rho}c ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = 0 (3.2)
{delta}t
Equation (3.2) simply expresses the conservation of heat energy. This is
most easily seen by integrating the equation over a fixed region, and using
Gauss's divergence theorem to change the volume integral of the divergence
(the leftmost term) into a surface integral. Then the equation just states
that whatever heat flows across the boundary must come from internal heart
generated (such as from metabolism within the human body or from
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(13)
HEAT FLOW IN CRYONIC SUSPENSION
change of phase), or from heat liberated from storage in the solid within
the contained region.
Substituting equation (3.1) into equation (3.2), we obtain:
{delta}T(r',t)
ś[kT(r't)] - g(r',t) + {rho}c ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = 0 (3.3)
{delta}t
We will make two simplifying assumptions to this general heat flow
equation:
A. We assume no heat generation within the solid, i.e. g(r',t) = 0.
Our model is designed to treat heat flow in the frozen human patient,
where heat flow from metabolism has already been arrested, and where
liberation of heat from freezing to the solid phase has already
occurred.
B. We assume the thermal conductivity k to be constant (i.e. independent
of temperature).
Then equation (3.3) simplifies to the usual heat flow equation (Fourier's
equation):
1 {delta}T(r',t)
żT(r't) = Ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = 0 (3.4)
ą {delta}t
For steady state heat flow, where the time derivative on the right side
vanishes, this reduces to Laplace's equation:
żT(r't) = 0 (3.5)
which is one of the most thoroughly investigated and solved equations in
all mathematical physics.
4. DIMENSIONLESS VARIABLES
It is natural and useful to express all equations in terms of
dimensionless variables and parameters. Such presentation simplifies the
equations, expresses the variables on a scale which is appropriate to the
problem, and permits the solution of diverse problems by use of standard
graphs and tables, simply remultiplying by the appropriate scale factors.
When possible, we will normalize variables to range between 0 and 1.
Dimensionless Position
We let:
r'
r'' = Ä (4.1)
L
-----------------------------------------------------------------------
(14)
HEAT FLOW IN CRYONIC SUSPENSION
where L is a characteristic linear dimension of the solid.
In simple geometries such as spheres or cylinders, we use the obvious
choice:
L = Radius [m] (4.2a)
In a solid of finite volume but arbitrary shape, we assign a characteristic
linear dimension as:
V
L = Ä [m] (4.2b)
A
In heat flow within an infinite or semi-infinite solid, there is no natural
standard length to use. Here, to continue presenting our equations in
terms of dimensionless variables, we arbitrarily assign:
L = 1 [m] (4.2c)
In this case r'' -> r (or in one dimensional heat flow x'' = x), in
2
magnitude but ignoring the dimension [m].
In each case, the mapping r -> r'' immediately induces mappings R ->R''
3
and S ->S'', which have corresponding dimensionless volume V'' and area A .
Note that when L is determined by (4.2b), we have:
3
A V
V = A = Ä (if L = Ä ) (4.3)
2 A
V
Having defined L, we immediately obtain the dimensionless gradient
operator:
= L (4.4)
Dimensionless Time
Dimensionless time is given by:
ąt
t'' = ÄÄ (4.5)
2
L
-----------------------------------------------------------------------
2 The precise choice of L in different solids varies with the
literature source; e.g. some sources use equation (4.2b) to determine L in
all solids. For a sphere, L from (4.2b) = 3 x L from (4.2a). For the
infinite cylinder, L from (4.2b) = 2 x L from (4.2a). Other normalizations
1/3
are possible, such as choosing L = V , which would make V = 1. Each
choice simplifies some equations at the expense of others.
3 For the infinite cylinder, which is really a two dimensional
problem, we only carry out the mapping in the radial direction and not in
the axial direction.
-----------------------------------------------------------------------
(15)
HEAT FLOW IN CRYONIC SUSPENSION
where t'' is also known as the Fourier number (F0), and contains the
constant of proportionality necessary to render (3.4) dimensionless. t''
measures the ratio of the average temperature of a slab to a constant
temperature difference maintained between its two faces, when heat flows
through one face separated from the opposite insulated face by a distance
L.
Other Dimensionless Variables
Of the remaining principal SI units, we have no need for a
dimensionless mass variable. Mass only appears in our equations in the
product {rho}c, from which it cancels out.
Dimensionless temperature is given in (5.3) below.
5. INITIAL AND BOUNDARY CONDITIONS
Solutions to equation (3.4) are subject to initial and boundary
conditions, which describe the initial temperature distribution within the
solid and on the boundary, and the rate of heat flow across the boundary.
In the real world, these conditions encountered can be perfectly
arbitrary. But in describing a particular physical configuration, they can
become too complex to permit a tractable analytical solution to the
problem.
For our purposes, we will restrict consideration to heat flow from an
instantaneous point source; across a plane; from a highly insulated
solid; from a sphere; or from a cylinder; under the following initial
and boundary conditions:
Initial Conditions
We let:
T = uniform initial temperature in R (K) (5.1)
0
T = uniform initial temperature on S (K) (5.2)
ģ
from which we define the dimensionless temperature scalar field:
T(r',t) - T
ģ
T''(r",t'') = ÄÄÄÄÄÄÄÄÄÄÄÄ (5.3)
T - T
0 ģ
4
Our initial conditions become:
T''(r",0) = 1 in R'' (5.4)
T''(r",0) = 0 on S'' (5.5)
-----------------------------------------------------------------------
4 Some sources, such as [4] in certain equations, adopt the reverse
convention by using the scalar (T'')' = 1 - T''.
-----------------------------------------------------------------------
(16)
HEAT FLOW IN CRYONIC SUSPENSION
Boundary Conditions
In the case of greatest interest, we consider cooling or warming the
human body with an external heat conducting fluid (liquid or gas). For
example, we may cool the body by surrounding it with liquid isopropyl
alcohol cooled by dry ice, or by liquid nitrogen, or in the vapor phase of
liquid nitrogen.
Let n" be an outward normal vector from the boundary surface. We
consider solutions to (3.4) under the very general boundary condition:
n"ś"T'' = -Bi''T'' on S'' for t'' > 0 (5.6)
If we let n'' measure dimensionless distance along the outward normal
vector, the left side of (5.6) reduces to {delta}T''/{delta}n''.
In the above equation,
Lh
Bi'' = ÄÄ is the Biot number (dimensionless) (5.7)
k
The Biot number measures the rate of heat convection at the surface divided
by the rate of heat conduction across the enclosed region.
In the cases treated in Section 7 and 11 where L is determined from
(4.2b), we will use the symbol BI'' for the Biot number, to distinguish it
from the value Bi'' that would be obtained from (4.2a).
A solution to the problem is required to satisfy the heat flow equation
(3.4) inside the region R'' for t'' > 0, and to approach the initial and
boundary conditions (5.4) - (5.6) as pointwise limits.
Special Cases of Boundary Conditions
A. In the base Bi'' ->ģ, wherein the heat flow at the boundary is very
large compared to the solid, (5.6) reduces to:
T(r',t) = T on S for t > 0 (5.8)
ģ
i.e., we maintain the boundary at a constant temperature.
B. The case Bi'' ->0 is that of rapid heat flow across the solid as
compared to convection at a highly insulate boundary. In the limit
Bi'' = 0, (5.6) reduces to:
n"śT = 0 on S for t > 0 (5.9)
Using (3.1) and (2.1), we have that:
F (r',t) = 0 on S for t > 0 (5.10)
n"
]-----------------------------------------------------------------------
(17)
HEAT FLOW IN CRYONIC SUSPENSION
so this limit represents a perfectly insulated boundary, across which
no heat flows. The temperature within the solid just remains at the
constant value T as in (5.1).
0
6. DIMENSIONLESS HEAT FLOW EQUATIONS
We can now define the dimensionless heat flux vector field by:
Lq'(r',t)
q"(r",t) = ÄÄÄÄÄÄÄÄÄÄ (6.1)
k(T - T )
0 ģ
from which we immediately obtain the total dimensionless planar heat flow
in the direction of unit vector n" as:
ōt
F' (r",t') = ³ n"śq"(r",s) d's' (6.2)
n" õ0
and define dimensionless heat generation by:
Lżg(r',t)
g(r",t) = ÄÄÄÄÄÄÄÄÄÄ (6.3)
k(T - T )
0 ģ
The dimensionless average temperatures in the region and on the surface are
give by:
1 ō ō ō
= Ä ³ ³ ³ T''(r",t)d''V'' (6.4)
R'' V''õ õ õR''
1 ō ō
= Ä ³ ³ T''(r",t)d''A'' (6.5)
S'' A''õ õS''
while the dimensionless heat energy contained within a region is given by:
Q(t)
Q''(t'') = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (6.6a)
3
{rho}cL (T - T )
0 ģ
= V'' from (2.5) (6.6b)
R''
so that the dimensionless heat content initially has the value V'' and over
time falls to 0 (if Bi'' > 0).
If length, time, temperature, and energy are measured in new units
5
defined by (4.2b), (4.5), (5.3), and (6.6a), then our fundamental
3 2
constants take on values V'' = A'' = A /V , k'' = {rho}c'' = ą" = L'' = 1,
and h'' = Bi''.
-----------------------------------------------------------------------
5 This choice of units dictates that if we had need for dimensionless
mass, it is given by:
2
ą m
m = (Ä) ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
L 3
{rho}cL (T - T )
0 ģ
-----------------------------------------------------------------------
(18)
HEAT FLOW IN CRYONIC SUSPENSION
Assuming the constants k, {rho}, and c are independent of temperature,
the basic heat flow equations (3.1), (3.2), (3.3), and (3.4) can be
expressed in terms of dimensionless variables as:
q" = -T'' (6.7)
{delta}T''
śq" - g'' + ÄÄÄÄÄÄÄÄÄÄ = 0 (6.8)
{delta}t''
{delta}T''
żT'' + g'' + ÄÄÄÄÄÄÄÄÄÄ (6.9)
{delta}t''
{delta}T''
żT'' = ÄÄÄÄÄÄÄÄÄÄ (7.0)
{delta}t''
7. GLOBAL REFORMULATION OF EQUATIONS
We can obtain additional insight into the differential heat flow
equation (6.10) and the differential boundary condition (5.6) by
reformulating them in global terms. Let (R'')' be an arbitrary subregion
of R'', and (S'')' be its surface. Integrating (6.10) over the region
(R'')', and applying Gauss's divergence theorem to the left side, we have:
ō ō ō ō ō {delta}T''
³ ³ n"śT''d''A'' = ³ ³ ³ ÄÄÄÄÄÄÄÄÄ d''V'' (7.1)
õ õ(S'')' õ õ õ(R'')' {delta}t''
Pulling the time derivative outside the integral and using the definitions
(6.4) and (6.5), we have:
d'' (A'')'
(R'')'
ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = ÄÄÄÄÄ for all (R'')' subset of R (7.2)
d''t'' (V'')' (S'')'
Since all of the above steps are reversible, (7.2) is an equivalent
formulation of (6.10).
Special Case: Heat Flow in the Whole Region
If we let (R'')' = R'' in (7.2), use (4.3) and substitute the boundary
condition (5.6), we obtain:
d''
(R'')'
ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = - B''I'' (7.3)
d''t'' (S'')'
so that the average temperature in the region decreases at a rate jointly
proportional to the Biot number and the average temperature on the
surface. In the presence of (7.2), this is an equivalent formulation of
(5.6).
-----------------------------------------------------------------------
(19)
HEAT FLOW IN CRYONIC SUSPENSION
Special Case: Heat Flow at a Point
If we contract the linear dimensions of the subregion toward 0 about a
point r", we obtain:
{delta}'' (A'')'
(R'')'
ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ = lim ÄÄÄÄÄ (7.4)
{delta}''t'' (V'')'/(A'')'->0 (V'')' (S'')'
which gives the time derivative of T'' in terms of the limiting value of
the net flux of T'' through a small surface about the point.
8. USEFUL MATHEMATICAL FUNCTIONS
We define several functions that will be used in the subsequent
solutions. First let
1 (x - ę)ż
N''(x''';ę,å) = ÄÄÄÄÄÄÄÄ exp (ÄÄÄÄÄÄÄÄÄÄÄÄ) (8.1)
________ 2åż
ū 2ćåż
be the normal (Gaussian) probability density function with mean ę and
variance åż.
The normal probability distribution function č is defined from N'' by
ōx''
č(x'') = ³ N''(u''; 0,1) d''u'' (8.2a)
õ-ģ
1 ōx' uż
= ÄÄÄÄ ³ exp (-ÄÄÄ) (8.2b)
____ õ-ģ 2
ū 2ć
which has values
č(-ģ) = 0, č(0) = .5, č(ģ) = 1 (8.3)
A close relative of č is the error function:
__
erf(x'') = 2č(ū 2 x'') - 1 (8.4a)
2 ōx''
= ÄÄÄÄ ³ exp(-u''ż)d''u'' (8.4b)
__ õ0
ū ć
which has the properties
erf(-x'') = -erf(x''), erf(0) = 0, erf(ģ) = 1 (8.5)
and from which we define the error function complement as
erfc(x'') = 1 - erf(x'') (8.6)
-----------------------------------------------------------------------
(20)
HEAT FLOW IN CRYONIC SUSPENSION
The error function can be represented by the following infinite series:
n''
2x'' ģ (2x''ż)
= ÄÄÄÄÄÄ exp(-x''ż) ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (8.7a)
__ n=0 1ś3śśś(2n''+1)
ū ć
2x'' 2x''ż
= ÄÄÄÄÄÄ exp(-x''ż) (1 + ÄÄÄÄÄ + śśś (8.7b)
__ 3
ū ć
useful for small values of x.
The asymptotic expansion of the error function complement as x'' ->ģ is
given by:
1 ģ 1ś3śśś(2n''-1)
erfc(x'') ~ ÄÄÄÄ exp(-x''ż) (1 + ä (-1) ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ) (8.8a)
__ n=1 n''
ū ć (2x''ż)
1 1
= ÄÄÄÄ exp(-x''ż) (1 - ÄÄÄÄÄ + śśś ) (8.8b)
__ 2x''ż
ū ć
The expansions presented in this form are particularly useful, since
solutions to the linear heat flow equation often contain the term
6
exp(xż) erf[c](x'') as in (10.2c)
We now temporarily relax strict rigor in favor of providing an
intuitively powerful tool for finding solutions. We first note that as å
becomes small, the mass of N'' as a function of x'' becomes highly spiked
and localized at the origin. This leads us to formally "define" the Dirac
7
delta function as:
{delta}(x'') = lim N''(x'';0,å) (8.9)
å->0
Clearly]
{delta}(x'') = 0 for x does not equal 0 (8.10)
-----------------------------------------------------------------------
6 cf. [1] pp. 297 & 298.
7 This function was "invented" by the physicist Paul Dirac in the late
1920's, as part of his reformulation of the basic laws of quantum
mechanics. There is no function which has the properties (8.10) and
(8.11d) within classical real analysis. The rigorously inadmissible step
above occurs in (8.11b) when the limit is passing through the integral
sign. But as long as we only use the properties of the delta function
while it appears as an integrand and limits are taken outside the integral,
all is well. Such use of the delta function can be rigorously justified
using the theory of distribution later developed by the mathematician
Laurent Schwartz.
-----------------------------------------------------------------------
(21)
HEAT FLOW IN CRYONIC SUSPENSION
while its value at the origin is infinite in such a way that:
ōģ ōģ
³ {delta}(u'')d''u'' = ³ lim N''(u''; 0,å)d''u'' (8.11a)
õ-ģ õ-ģ å->0
ōģ
= lim ³ N''(u''; 0,å)d''u'' (8.11b)
å->0 õ-ģ
= lim 1 (8.11c)
å->0
= 1 (8.11d)
The delta function can thus be used to represent a point source of unit
heat at the origin. More generally, we can formally represent an arbitrary
function f'' as a superposition of point sources:
ōģ
f''(x'') = ³ f''(u''){delta}(u'' - x'')d''u'' (8.12)
õ-ģ
- the integral just picks out the value of f'' where all the mass of
{delta} is concentrated.
We will now present various solutions to the dimensionless heat flow
equation (6.10).
9. BASIC SOLUTION: HEART FLOW IN ONE DIMENSION
In the most elementary case, we consider heat flow from a very thin
flat plate of infinite extent in the y'' and z'' directions. We let one
(dimensionless) unit of heat per unit area be placed in the plane x'' = 0
at time t'' = 0. Since this source of heat is (ideally) compressed into a
plane of zero thickness, this initial temperature distribution can be
represented as:
T(x'',0) = {delta}(x'') (9.1)
Then the solution to (6.10) is:
____
T(x'',t'') = N''(x''; 0, ū 2t'') (9.2a)
1 x''ż
= ÄÄÄÄÄÄÄÄ exp(- ÄÄÄÄ) (9.2b)
______ 4t''
ū 4ćt''
as can be verified directly by differentiation of this solution. It is
also immediate from the definition (8.9) that N'' satisfies the initial
condition (9.1).
Now in the general case, on the x'' axis we are given an arbitrary
initial distribution T''(x'',0), -ģ < x'' < ģ. We use (8.12) to represent
T'' as a superposition of point sources:
-----------------------------------------------------------------------
(22)
HEAT FLOW IN CRYONIC SUSPENSION
ōģ
T''(x'',0) = ³ T''(u'',0){delta}(u'' - x'')d''u'' (9.3)
õ-ģ
Since equation (6.10) is linear in the temperature T'', the same
superposition of elementary solutions will be the solution to the general
8
case:
1 ōģ (x'' - u'')ż
T''(x'',t'') = ÄÄÄÄÄÄÄÄ ³ T''(u'',0) exp(- ÄÄÄÄÄÄÄÄÄÄÄÄ)d''u'' (9.4)
______ õ-ģ 4t''
ū 4ćt''
where again it is immediate from (8.9) that T'' satisfies the initial
condition (9.3).
10. HEAT FLOW IN A SEMI-INFINITE SOLID
We consider heat flow across the plane surface x'' = 0 from the region
x > 0. The solution to this problem is particularly useful, since
virtually any surface we naturally encounter can be approximated by this
planar model, if we confine the approximation to small enough values of x''
and t''. We use (4.2c) to determine L and thus Bi''.
The solution to (6.10) under the initial and boundary conditions
9
(5.4) - (5.6) is:
x'' x''ż 2Bi''t'' ż x'' 2Bi''t''
T''(x'',t'')=erf(ÄÄÄ)+exp(ÄÄ ((1+ÄÄÄÄÄÄÄÄ)-1))erfc(ÄÄÄ (1+ÄÄÄÄÄÄÄ)) (10.1a)
___ 4t'' x'' __ x''
ū4t'' ū4t''
x'' x'' ____
=erf(ÄÄÄ)+exp(Bi''x'' + Bi''żt'') erfc(ÄÄÄ + Bi ū t'' (10.1b)
___ __
ū4t'' ū4t''
This solution contains three parameters x'', t'', Bi'', but it may be
____
expressed in terms of any two of the combinations x/ū 4t', Bi''x'', and
___
Bi''ūt'', useful for graphing the solution.
The heat flow across the boundary per unit area and time in the -x''
direction is given by:
{delta}T''³
q'' (0,t'') = ÄÄÄÄÄÄÄÄÄ ³ from (6.7) (10.2a)
-x'' {delta}x''³ x'' = 0
= Bi''T''(0,t'') from (5.6) with n'' = -x'' (10.2b)
___
= Bi''exp(Bi''żt'')erfc(Bi''ūt'' from (10.1b) (10.2c)
Integrating this equation with respect to t'', we find from (6.2) that the
total heat flow across unit area in the -x'' direction is:
1 ___ 4t'' 1/2
F'' (0,t') = ÄÄÄÄ [exp(Bi''żt'')erfc(Bi''ūt'') -1] + (ÄÄÄÄ) (10.3)
-x'' Bi'' ć
-----------------------------------------------------------------------
8 cf. [8] pg. 45.
9 cf. [4] pg. 34 or [7] pg. 177.
-----------------------------------------------------------------------
(23)
HEAT FLOW IN CRYONIC SUSPENSION
Special Case: Constant Boundary Temperature
If we let Bi'' ->ģ, which entails that the boundary condition (5.6)
becomes:
T''(0,t'') = 0 for t'' > 0 (10.4)
-- the boundary is kept at a constant temperature -- then the solution
(10.1b) simplifies to:
x''
T''(x'',t'') = erf(ÄÄÄÄ) (10.5)
___
ū4t''
From (10.2a) we then obtain:
1
q'' (0,t'') = ÄÄÄÄ (10.6)
-x'' ___
ūćt''
and using (6.2) we have:
4t'' 1/2
F'' (0,t'') = (ÄÄÄÄÄ) (10.7)
-x'' ć
The last two equations could, of course, also have been obtained from
(10.2c) and (10.3) by use of the expansion (8.8b).
Special Case: Highly Insulated Boundary
If we let Bi'' ->0, we describe the case where the heat flow across the
solid is very rapid as compared to convection at its highly insulated
boundary. If we carry out the Taylor series expansion of (10.1a) as a
function of the parameter Bi'', then to first order in Bi'' we obtain:
2
4t''1/2 x'' x'' x''
T''(x'',t'')=1-Bi''((ÄÄÄ) exp(- ÄÄÄÄ)-x erfc(ÄÄÄÄ)) for Bi'' <<ÄÄ (10.8)
ć 4t'' ___ t''
ū4t''
From *10.2c) using (8.7b), we see that:
1
q'' (0,t'') = Bi'' for Bi'' <<ÄÄÄ (10.9)
-x'' __
ūt''
and using (6.2),
1
F'' (0,t'') = Bi'' for Bi'' <<ÄÄÄ (10.10)
-x'' __
ūt''
so that the total heat flow across the boundary increases linearly with
time.
-----------------------------------------------------------------------
(24)
HEAT FLOW IN CRYONIC SUSPENSION
11. HEAT FLOW FROM A HIGHLY INSULATED SOLID
We consider the case of a solid of arbitrary shape in which heat
conduction across the solid medium is very rapid as compared to heat
convection at the boundary. Then the temperature within the solid will be
nearly the same at all positions. In this case, we use (4.2b) to define L,
and consequently BI''. For most solids, if BI'' < .1 then the temperature
10
within the solid can be taken as uniform to within a 5% error.
We now allow for the possibility that the specific heat c is not a
constant, but depends linearly upon temperature. This is approximately
true of ice, whose heat flow characteristics can be used as a first
11
approximation to the frozen human body. We assume
c = c (1 + b''T'') (11.1)
ģ
In (4.5) we use ą = k/{rho}c , so that instead of (6.10) we obtain:
ģ ģ
2
2 {delta}T'' {delta}T'' b'' {delta}T''
T'' = (1 + b''T'')ÄÄÄÄÄÄÄÄ = ÄÄÄÄÄÄÄÄ + Ä ÄÄÄÄÄÄÄÄÄ (11.2)
{delta}t'' {delta}t'' 2 {delta}t''
By precisely the same argument as was used to obtain (7.3),
2
d'' d''
R'' b'' R''
ÄÄÄÄÄÄÄÄÄ + Ä ÄÄÄÄÄÄÄÄÄÄ = -B''I'' (11.3)
d''t'' 2 d''t'' S''
But since the temperature has a nearly uniform value T''(t'') within R'',
÷ ÷ T''
R'' S''
2 2
÷ T'' (11.4)
so that (11.3) becomes:
d''T''
(1 + b''T'') ÄÄÄÄÄ = -BI'' T'' (11.5)
d''t''
d''T''
(1/T'' + b''T'') ÄÄÄÄÄ = -BI'' (11.6)
d''t''
Integrating (11.6) from time 0 to t, we obtain:
log(T'') + b''(T'' - 1) = - BI'' t'' (11.7)
-----------------------------------------------------------------------
10 [7] pg. 140
11 see data in [12] pg. D-138; also [2] Chapter 29.
-----------------------------------------------------------------------
(25)
HEAT FLOW IN CRYONIC SUSPENSION
This equation cannot be solved in closed form for T'', but T'' can be
obtained by Newton's method. We let
T = 1 (11.8)
0
log(T'' ) + b''(T'' - 1) + BI'' t''
n'' n''
T = T - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (11.9)
n''+1 1/T'' + b''
n''
and iterate until the T'' 's converge to the desired degree of precision.
n''
Special Case: Constant Specific heat
If we let b'' = 0, we can obtain T'' explicitly from (11.7):
T'' = exp(- BI'' t'') (11.10)
This result could have been seen directly from (7.3). This special case is
12
known in the literature as the homogeneous billet.
12. HEAT FLOW FROM A SPHERE
The sphere can be used as a model of heat flow through the head, either
during whole body suspension or during neurosuspension. Under the uniform
initial and boundary conditions we have adopted, all heat flow must be in
the radial direction. We use (4.2a) to determine L, and thus Bi''.
The solution to (6.10) under the initial and boundary conditions (5.4)
13
- (5.6) is:
2
sin(R'' r'') exp(-R'' t'')
ģ n'' n''
T''(r'',t'') = 2 ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (12.1)
n''=1 (R'' r'')[R'' sin(R'' )/Bi''-cos(R'' )]
n'' n'' n'' n''
where R , R , ś ś ś are the solutions to the eigenvalue equation:
1 2
R'' cot(R ) + Bi'' - 1 = 0 (12.2)
n'' n''
The denominator in (12.1) can be reformulated in various ways, using
the eigenvalue equation (12.2), so that the expression does not contain
indeterminate forms (0/0) when r'' and Bi'' approach desired limiting
values. We have written (12.1) so that it converges to the proper limit as
Bi'' ->ģ.
The solution at r'' = 0 is obtained by taking the limit of (12.1) as
r'' ->0:
2
exp(-R'' t'')
ģ n''
T''(0,t'') = 2 ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (12.3)
n''=1 R'' sin(R'' )/Bi''-cos(R'' )
n'' n'' n''
-----------------------------------------------------------------------
12 [5] pg. 139 and [7] pg. 140.
13 cf. [4] pg. 91.
-----------------------------------------------------------------------
(26)
HEAT FLOW IN CRYONIC SUSPENSION
The average temperature within the sphere is given by:
2
exp(-R'' t'')
ģ n''
= 6 ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (12.4)
n''=1 2 2
R'' (1-1/Bi''+(R'' /Bi'') )
n'' n''
Special Case: Constant Boundary Temperature
If we let Bi'' ->ģ, the eigenvalue equation (12.2) simplifies to:
tan(R'' )
n''
ÄÄÄÄÄÄÄÄÄ = 0 (12.5)
R''
n''
whose solutions are:
R'' = n''ć for 0 < n'' < ģ (12.6)
n''
The solution (12.1) simplifies to:
2
ģ n''+1 sin(n''ćr'')exp(-(n''ć) t'')
T''(r'',t'') = 2 ä (-1) ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (12.7)
n''=1 n''ćr''
The solution at r'' = 0 is again obtained by taking the limit as
r''->0:
ģ n''+1 2
T''(0,t'') = 2 ä (-1) exp(-(n''ć) t'') (12.8)
n''=1
The average temperature within the sphere is given by:
2
ģ exp(-(n''ć) t'')
= 6 ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (12.9)
n''=1 2
(r''ć)
Special Case: Highly Insulated Boundary
If we let Bi'' ->0, we describe the case where heat flow across the
sphere is very rapid as compared to convection at its highly insulated
boundary. In this limit, the first solution to the eigenvalue equation
(12.2) becomes approximately:
______
R'' = ū 3Bi'' (12.10)
1
In this case, the nearly uniform temperature within the sphere can be
obtained from the first term in (12.4):
= exp(-3Bi''t'') (12.11)
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HEAT FLOW IN CRYONIC SUSPENSION
We note that if we use (4.2b) rather than (4.2a) to define the
characteristic linear dimension of the sphere, then with the revised value
of t'' and Bi'', (I12.11) becomes:
= exp(-Bi''t'' ) (12.11)
BI'' BI''
which was the solution obtained in (11.10).
13. HEAT FLOW FROM A CYLINDER
2
J''(R'' r'')exp(-R'' t'')
ģ 0 n'' n''
T''(r'',t'') = 2 ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (13.1)
n''=1 2
R'' J''(R'' )[1+(R'' /Bi'') ]
n'' 1 n'' n''
Here J'' and J'' are the Bessel functions of order 0 and 1, where R , R ,
0 1 1 2
ś ś ś are the solutions to the eigenvalue equation:
R'' J'' (R'' )-Bi'' J'' (R'' ) = 0 (13.2)
n'' 1 r'' 0 n''
and we have again used the eigenvalue equation to formulate the solution so
that it converges to the proper limit as Bi'' ->ģ.
The average temperature within the cylinder is given by:
2
exp(-R'' t'')
ģ n''
= 4 ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (13.3)
n''=1 2 2
R'' [1+(R'' /Bi'') ]
n'' n''
Special Case: Constant Boundary Temperature
If we let Bi'' ->ģ, the eigenvalue equation (13.2) simplifies to:
J'' (R'' ) = 0 (13.4)
0 n''
15
The roots of (13.4) are widely tabulated.
In this limit the solution (13.1) becomes:
2
J''(R'' r'')exp(-R'' t'')
ģ 0 n'' n''
T''(r'',t'') = 2 ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (13.5)
n''=1 R'' J''(R'' )
n'' 1 n''
-----------------------------------------------------------------------
14 cf. [4] pg. 73, [8] pg. 102, and using (13.2).
-----------------------------------------------------------------------
(28)
HEAT FLOW IN CRYONIC SUSPENSION
The average temperature within the cylinder is:
2
exp(-R'' t'')
ģ n''
= 4 ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (13.6)
n''=1 2
R''
n''
Special Case: Highly Insulated Boundary
If we let Bi'' ->0, just as with the sphere we describe the case where
the heat flow across the cylinder is very rapid as compared to convection
at its highly insulated boundary. In this limit, the first solution to the
eigenvalue equation (13.2) becomes approximately:
______
R'' = ū 2Bi'' (13.7)
1
We obtain the nearly uniform temperature within the cylinder from the first
term in (13.3):
= exp(-2 Bi''t'') (13.8)
Again we note that if we use (4.2b) rather than (4.2a) to define the
characteristic linear dimension L of the cylinder, then with the revised
values of t'' and Bi'', (13.8) becomes:
= exp(-BI''t'' ) (13.9)
BI'' BI''
which was the solution obtained in (11.10).
14. ANALOGY BETWEEN HEAT FLOW AND DIFFUSION
The basic laws of heat flow (3.1) - (3.4) can be reinterpreted to
describe the process of diffusion within the human body. To describe
diffusion, such as of glycerol or DMSO during cryoprotective perfusion of
the human body, we require new fields such as:
3
C(r',t) = concentration of cryoprotectant [kg/m ] (14.1)
Here, matter father than heat is moving. So the isomorphism is
accomplished as follows, using the diffusion symbols of [4]:
-----------------------------------------------------------------------
15 for example, in [1], with shorter tables in [3], [4], [8].
-----------------------------------------------------------------------
(29)
HEAT FLOW IN CRYONIC SUSPENSION
HEAT FLOW <--> DIFFUSION
Quantity Unit Quantity Unit
3 3
{rho}cT J/m C kg/m
2 2
ą m /s D m /s
2 2
q' J/(m s) j' kg/(m s)
2 2
F J/m M kg/m
Q J m kg
h/{rho}c m/s ą m/s
Equation Name Equation Name
q' = -kT Fourier's law j'=-DC Fick's first law
2 1 {delta}T
T = Ä ÄÄÄÄÄÄÄÄ 2 1 {delta}C
ą {delta}t Fourier's equation C=Ä ÄÄÄÄÄÄÄÄ Fick's
D {delta}c second law
{delta}T
kÄÄÄÄÄÄÄÄ = -h (T - T ) boundary condition {delta}C
{delta}t ģ DÄÄÄÄÄÄÄÄ= -ą(C-C ) boundary
{delta}t ģ condition
15. REFERENCES
The list of references below includes several relevant texts that were
not explicitly referenced in the article.
[1] Abramowitz, M. and Stegun, I. ed., "Handbook of Mathematical
Functions," Dover, 1965.
[2] "ASHRAE Handbook & Product Directory," American Society of Heat
Refrigeration and Air Conditioning Engineers, 1977 Fundamentals.
[3] Carslaw, J., "Conduction of Heat in Solids, 2nd ed.," Oxford, 1959.
[4] Crank, J., "Mathematics of Diffusion," Oxford, 1956.
[5] Eckert, J. and Drake, R., "Analysis of Heat and Mass Transfer," McGraw
-Hill, 1972.
[6] Gray, D. ed,. "American Institute of Physics Handbook, 3rd ed.,"
McGraw-Hill, 1972.
[7] Kreith, F., "Principles of Heat Transfer, 3rd ed.," Harper & Row,
1973.
[8] Ozisik, M., "Heat Conduction," John Wiley & Sons, 1980.
-----------------------------------------------------------------------
(30)
HEAT FLOW IN CRYONIC SUSPENSION
[9] Quaife, A., "Mathematical Models of Perfusion Processes," "Manrise
Technical Review, 2:28-75, 1972.
[10] Sears, F. and Zemansky, M., "University Physics, 2nd ed.," Addison
-Wesley, 1955.
[11] Timmerhaus, K. ed., "Advances in Cryogenic Engineering," Vol. 8 pg.
267, Plenum Press, 1963.
[12] Weast, R. ed, "Handbook of Chemistry and Physics, 54th ed.," CRC
Press, 1973.
-----------------------------------------------------------------------
(31)
** PHOTOS **
** CAPTIONS --
"Hugh Hixon preparing to mount the shock absorbers on the dewar feet."
**
"Mounting the A-2542 dewar on its plywood base."
**
-----------------------------------------------------------------------
(32)
** PHOTOS **
** CAPTIONS --
"Picking up the dewar on its new base."
**
"Replacing the top of the vault, using the forklift/crane. Jerry Leaf at
the controls, Scott Greene and Mike Darwin guiding."
**
-----------------------------------------------------------------------
(33)
very well at running around and shouting a
lot. But, the real star of the day was ** PHOTO SPACE **
Hugh Hixon, who's thoughtful planning and ** CAPTION --
ingenious engineering design resulted in a
flawless operation from start to finish. "The forklift/crane.
Things went so smoothly we even had time to (And Brenda Peters,
intersperse moving operations with enter- our photographer.)"
taining some guests: ALCOR member Saul
Kent showed up with a couple of people from **
Florida, and we all paused for a little
talk and some cool refreshments.
On Monday, the 29th of July, the pat-
ients were returned to the dewar and boil-
off studies were begun to thoroughly eval-
uate the health of the A-2542. Over the
past few years a precise measurement of
the performance of the A-2542 has been all
but impossible. This was because there was
a tremendous amount of research material in
the dewar, much of it irregularly shaped
and some of it projecting above the liquid
*** TYPIST NOTE: ORIGINALLY THIS SPACE CONTAINED A GRAPH (A-2542 BOILOFF)
OF LN2 DEPTH IN INCHES VS. TIME IN HOURS. THIS GRAPH WAS EXPLAINED IN THE
FOLLOWING CAPTION. ***
A-2542 Boiloff. LN2 depth is measured as distance of liquid
surface from the top of the neck tube. Each inch of depth
equals 19.58 liters of LN2. Note that as the interval
between measurements gets longer, the boiloff rate decreases.
-----------------------------------------------------------------------
(34)
line. Also, since the dewar is being used not only for patient care, but
for research as well, small amounts of liquid are occasionally removed and
the lid is opened with some frequency. This prevents the careful
measurements which are necessary to establish dewar performance from being
taken. Thus, it was extremely important to get a baseline on the
container's performance.
As you can see from the accompanying graph, the A-2542 is performing
superbly. This latest series of measurements indicates that it is boiling
off 3.5 liters per day. The dewar is rated from the manufacturer to boil
off 3.4 to 3.8 liters per day, so it's right where it should be! This was
a little surprising since our initial evaluation four years ago, when it
first arrived, was that it was boiling off 4.5 liters per day. We have now
learned the importance of a longer evaluation period and the use of a
single individual employing a highly repeatable measuring technique.
All in all, moving into the vault went much more smoothly than expected
and we are all overjoyed and relieved to have this big project behind us
and to know that our patients are finally getting the kind of protection
they both need and deserve!
LETTER TO THE EDITORS:
Dear Mike,
"Cocoon," which you reviewed in the August issue of CRYONICS, is indeed
a wonderful film which stresses the immortalist philosophy. However, I
have some cautions for optimistic readers who may think this heralds the
dawn of a cryonics boom. We as cryonicists are hypersensitive to the
immortalist ideas expressed in the film and give them greater importance
than may most of the general public. Many of my friends here in
Indianapolis have seen "Cocoon" and have praised it highly -- yet only one,
even with my prompting, recognized the philosophy stated in the film as
being "immortalist" or having anything to do with my cryonics involvement.
And this is from people who have heard immortalist ideas from me. Most
people are viewing the film as pure entertainment. When pressed for a
"meaning," the best these people can do is "Old people are a lot livelier
than most young people understand" or "It would be great for the aliens to
invite us out into space with them." This may be the first great
immortalist film, but it won't lead anyone to cryonics.
. . . UNLESS we make the connection for them. Here is my suggestion.
When you are talking with people about the film, ask a group "If those
aliens landed now and offered you the same conditions as they offered the
old folks in the film -- that is, you would lead productive lives and learn
many of the secrets of the universe, you would never get older, you would
never be sick, and you wouldn't ever die; but you would have to leave the
time, the place, and the people (except for those coming along with you)
you know -- how many of you would go?"
You are likely to get several yes's. If so, you can discuss the idea of
living and learning forever in space. And eventually you can make the
observation that the aliens probably are not coming for us. If we want
that kind of life, it will have to be a "do-it-ourselves" proposition. And
the
-----------------------------------------------------------------------
(35)
opportunity for "do-it-ourselves" is already here in cryonics.
Don't ever think that movies like "Cocoon" or books like "Jitterbug
Perfume" (reviewed in June issue of CRYONICS) will do our work for us.
They merely provide us with a starting point and a climate from which to
launch an attempt to change individuals' ideas about immortality; to show
them that immortality is not just an entertaining plot device or vague
human dream, but that it is a potential reality which can affect their own
lives.
Stephen Bridge
Indianapolis, IN
THE SCANNING TUNNELING MICROSCOPE:
THE DOOR INTO MOLECULAR TECHNOLOGY
by Hugh Hixon
Over approximately the past year, the scientific press and journals
have been reporting the development and operation of a fundamentally new
scientific instrument. Able to resolve distances of the order of one one-
hundredth of an atomic diameter, the Scanning Tunneling Microscope (STM)
promises to be the precursor of a group of revolutions so far-reaching as
to defy accurate prediction.
For cryonics and life extension, what it appears to promise is a
particularly rapid entry into the molecular technology (MT) necessary to
discover the basic mechanisms of aging within the living cell, and to
perform the reconstructive work necessary to perform a reanimation from
cryonic suspension.
Among the earliest views made of what has up to now been a world beyond
anything but our mental conceptions are atom-by-atom maps of the surfaces
of crystals and components of one of nature's own molecular machines, the T-
4 bacteriophage. The pictures of the T-4 phage pieces are not easily
reconciled with our ideas of them gained from conventional transmission
electron microscopy, but this is a problem that will be overcome with
developments in experimental technique. The pictures of the crystal
surface, however, are unambiguous, so far as our present knowledge of such
things goes.
Both conceptually and practically, the heart of the STM is the essence
of simplicity. A three-axis drive moves the tip of a needle over the
surface of interest, at a distance so small that electrons from the surface
bridge the gap by quantum-mechanical tunneling. The resulting current,
easily detected by an instrument no more sensitive than a pH meter, is used
to control the distance of the needle above the surface, and that control
signal is also used to trace a picture of the surface.
What is not simple is getting used to the idea that a device
centimeters (10-2 meters) in size can regulate its position with an
accuracy of hundredths of Angstrom units (10-12 meters) a ratio of ten
billion to one! By comparison
-----------------------------------------------------------------------
(36)
you can think of using a continuous strip-mining machine (102's of meters)
to observe the components of a microelectronic chip (10-6 meters), and
still fall short by a factor of 100! The history of the invention has not
yet been written, but it will be interesting to read of how the IBM
researchers who created the STM overcame the mental barriers to the
construction of a device, all of whose components were available by 1900,
and whose underlying theory of quantum-mechanical tunneling was defined in
the early 1930's.
The STM's inventors are Gerd Binnig and Heinrich Rohrer of IBM's Zurich
Research Laboratory. In the August, 1985 issue of Scientific American,
they give a brief account of how the STM works, and some good pictures and
diagrams of their pioneering instrument. For a description of how the STM
works, I quote from Conrad Schneiker's forthcoming paper, NanoTechnology:
. . . Roughly speaking, STM's operate by scanning an extremely sharp,
electrically conducting needle tip within a few atomic radii of a
surface to be imaged. Variations in the spacing between the object
being "looked at" and the tip of a few tenths of nanometers (=
Angstroms, = 10-10 meters) result in observed (quantum vacuum
tunneling) current changes of over four orders of magnitude. This
permits atomic scale feature resolution under favorable conditions. In
addition, THE STM WILL WORK IN AIR, WATER, AND OIL WITH SOME LOSS OF
RESOLUTION. (Emphasis added).
Binnig and
Rohrer's list of co-
authored publications
is growing almost dai-
ly, as researchers in
every field that does
microscopic work move
to assess the potential
of the STM. As of
August, 1985, Schneiker
lists 32 papers on STM
in a draft of his
forthcoming paper,
NanoTechnology. STM's *** TYPIST NOTE: ORIGINALLY THIS SPACE
are being built all CONTAINED A LINE-DRAWING DIAGRAM OF THE
over the world, and SCANNING-TUNNELING MICROSCOPE, DESCRIBED IN
they are simple enough THE FOLLOWING CAPTION. ***
that it is likely that
there will be at least
one STM project in the
1986 Science Fair
(Westinghouse Science The STM. The sample "S" is mounted on the posit-
Talent Search, which ioning platform "L", and moved close to the scan-
promotes science ning needle. The needle tip is scanned across the
projects at the high surface of the sample by the 3-axis X, Y, and Z
school level). drives. Inset (b) diagrams the needle and surface
on a microscopic scale. Inset (a) diagrams the
In detail, the STM needle tip and surface on an atomic scale.
consists of: a posit-
ioner on piezoelectrically driven legs to move the object to be observed
close to the tip without smashing into it; the three-axis piezodrive,
which scans the
-----------------------------------------------------------------------
(37)
sensing tip over the object in a TV-like raster pattern, and moves it back
and forth to follow the surface of the object; the tip, which picks up the
electrons emitted from the object. (Depending on the application, the tip
may be either ion-milled, where the tip is first sharpened conventionally,
and the final sharpening is done by blasting away metal atoms with a
directed stream of ions, or simply used after mechanical sharpening, since
the sharpening process leaves fine microwires of material projecting from
the tip.); the current detector, which measures the quantum mechanical
tunneling current between tip and object; and which is connected to a
controller, a simple computer which keeps the current constant by varying
the voltage to the piezoelement controlling the distance between the tip
and object (Z-axis positioner), and which also controls the position of the
tip (X- and Y-axis positioners); the shock absorption system, which can be
as complex as Binnig and Rohrer's two-stage, magnetically damped, vacuum-
isolated workstage, or as simple as a stack of metal plates separated by
rubber grommets; and finally, a theory of operation, in this case the
quantum-mechanical prediction that electrons may be found away from a
surface (although at this scale, the concept of a surface becomes rather
uncertain).
As an adjunct to their tip sharpening, Binnig and Rohrer found that,
with proper technique, they could move a few atoms, or even a single atom
from the object to the tip, to make a final point on a scanning needle that
is only one atom in diameter! Not only does this give the scanning needle
the ultimate sharpness, but it is an immediate demonstration of the
possibility of moving single atoms around; i.e. -- molecular engineering.
It seems likely that practical engineering tips will be more complex,
incorporating molecules like B-12 on the tip (Vitamin B-12 is a complex
macrocyclic ring structure with an atom of cobalt in the center. Living
systems use it for one-carbon transfers in molecular synthesis).
Researchers have already begun examining DNA molecules under the STM.
While we have not seen pictures of the results, it is obvious that a very
early goal of this technology will be a DNA reader, able to read a strand
of DNA like a piece of punched teletype tape. A somewhat more complex goal
will be a molecular machine that can also write, or erase and rewrite, DNA.
In NanoTechnology, Conrad Schneiker traces the concept of molecular
machines to a speech and paper by physicist Richard Feynman in 1959,
outlining a system to produce more and more, smaller and smaller machines.
(This reprint is available from ALCOR as part of the package on the
Scientific Basis of Cryonics.) It now appears that Feynman's route to
molecular devices, making them smaller in stepwise fashion, has been short-
circuited. As Schneiker points out, the STM appears to be able to go from
something that can be held in one's hand to the smallest machines
conceivable, in one step. Feynman is reportedly delighted. Readers of
science fiction may attempt a previous credit with Robert Heinlein's 1940's
story, Waldo, but such a claim seems tenuous. Heinlein used the idea of
building smaller and smaller machines only once, and in passing.
To further the development of STM's and molecular machines, Conrad
Schneiker has proposed and will fund a series of competitive challenges and
prizes to the engineering and science community at large, for progressively
smaller STM's. More information on this competition is available in his
ongoing paper, NanoTechnology, available from ALCOR for $4.00, in its
latest iteration.
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(38)
SEPTEMBER-NOVEMBER 1985 MEETING CALENDAR
ALCOR meetings are usually held on the
first Sunday of the month. Guests are
welcome. Unless otherwise noted, meet-
ings start at 1:00 PM. For meeting
directions, or if you get lost, call
ALCOR at (714) 738-5569 and page the
technician on call.
-----------------------------------------------------------------------
The SEPTEMBER meeting will be at the home of:
(SUN, 8 SEPT 1985) Mike Darwin and Scott Greene
(SECOND SUNDAY) 350 W. Imperial Highway, #21
Brea, CA
Tel: (714) 990-6551
DIRECTIONS: Take the Orange Freeway (Hwy 57) to Imperial Highway (Hwy 90),
and go west through Brea on Imperial Highway. 350 is about
one mile from the freeway, and in the third block beyond Brea
Blvd., on the south (left) side. If the gate is closed, park
on the streets north of Imperial. Be careful crossing
Imperial. There is a blind curve to the east and a blind hill
to the west at this point.
-----------------------------------------------------------------------
The OCTOBER meeting will be at the home of:
(SUN, 6 OCT 1985) Paul Genteman
535 S. Alexandria, #325
Los Angeles, CA
DIRECTIONS: From the Santa Monica Freeway (Interstate 10), exit at Vermont
Avenue, and go north to 6th St.
From the Hollywood Freeway (US 101), exit at Vermont Avenue,
and go south to 6th St.
Go west on 6th 4 blocks to Alexandria, and turn right. 535 is
the first apartment building on the west side of the street.
Ring #325 and someone will come down to let you in.
-----------------------------------------------------------------------
The NOVEMBER meeting will be at the home of:
(SUN, 3 NOV 1985) Maureen Genteman
524 Raymond Avenue, #12
Santa Monica, CA
DIRECTIONS: Take the Santa Monica Freeway (Interstate 10) to Santa Monica
and get off at the 4th Street exit. Turn south (left) on
4th. Go south on 4th to Ocean Park Ave. (4-way flashing
stop). Go left on Ocean Park, down ramp to stop and up to 6th
St. on Ocean Park. Turn right on 6th. Raymond is the second
street. Turn right on Raymond. 524 is on the left. #12 is
on the second floor.