the effects of complexity
TRANSCRIPT
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Nuclear Engineering and Design 204 (2001) 127
The effects of complexity, of simplicity and of scaling inthermal-hydraulics
Novak Zuber *
703 New Mark Esplanade, Rock6ille, MD 20850, USA
Received 22 May 2000; accepted 22 May 2000
Abstract
This lecture has a twofold purpose. First, we will assess the state of the art and the trends in thermal-hydraul
(T-H) technology, within the context of replicating and non-replicating information systems. Four T-H examples a
used to illustrate that an ever-increasing complexity in formulating and analyzing problems leads to inefficien
obsolescence and evolutionary failure. By contrast, simplicity, which allows for parsimony, synthesis and clarity
information, ensures efficiency, survival and replication. This comparison (complexity versus simplicity) also provid
the requirements and guidance for a success path in T-H development. The second objective of this paper is
demonstrate that scaling provides the means to process information in an efficient manner, as required by competiti
(and, thereby, replicating) systems. To this end, the lecture summarizes the essential features of the Fractio
Change, Scaling and Analysis approach, which offers a general paradigm for quantifying the effects that an agentchange has on a given information system. The paper will further demonstrate that a single concept and a sin
method may be used to scale and analyze all transport processes in a given field of interest (fluid mechanics, he
transfer, etc.) and/or across fields and disciplines (mechanics, biology, etc.) Therefore, the paradigm: (1) ensu
economy and efficiency in addressing and resolving technical or scientific problems; and (2) enables a cultu
cross-pollination between different information systems (disciplines). By means of a simple example in the Append
we shall: (1) demonstrate the efficiency to be gained through scaling; and (2) illustrate the inefficiency a
wastefulness of computer-based safety studies as presently conducted. 2001 Elsevier Science B.V. All rig
reserved.
www.elsevier.com/locate/nuceng
1. Introduction
1.1. Purpose
I would like to begin my remarks by thanking
the Organizing Committee for inviting me to par-
ticipate. At this stage of my life, I am not certain
how many more opportunities I shall have
comment on the state of the art as concerthermal-hydraulics (T-H) technology, and to ou
line avenues that, in my opinion, hold gre
promise for the improvement, advancement a
enrichment (by cultural cross-pollination) of t
and other branches of technology and of scien
This paper has, therefore, a twofold purpo
First, I shall assess the developmental trends * Tel.: +1-301-4243585.
0029-5493/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 2 9 - 5 4 9 3 ( 0 0 ) 0 0 3 2 4 - 1
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the T-H technology and register my concerns with
regard to the dangers that lie ahead if the ap-
proach to formulating and analyzing T-H prob-
lems continues to increase in complexity (with no
greater degree of efficiency, I might add). I am
well aware that my views on this subject will be ill
received by some code developers and users, and
certainly by most of the code jockeys. So be it.
Were I to give you less than my candid opinion,
based on 48 years of experience in this technology
(starting with my first year as a graduate student
at UCLA), I would be remiss in my responsibili-
ties as a member of two professional societies, i.e.
the American Nuclear Society (ANS) and the
American Society of Mechanical Engineers
(ASME).
My second objective is to summarize and
demonstrate the key features of the FractionalChange, Scaling and Analysis method (FCSA),
which are: simplicity, parsimony, synthesis, effi-
ciency and versatility.
To this end, this paper will demonstrate that a
single concept (simplicity and parsimony) and a
single methodology (again, simplicity and parsi-
mony) may be used for the following.
1. To scale all transfer processes associated with
particles, waves, diffusion and vorticity (syn-
thesis) across hierarchical levels ranging from
Kolmogorovs micro scale to a nuclear reactor(synthesis and efficiency).
2. To derive Kolmogorovs scaling relations for:
the inertial subrange; and
the micro range (synthesis).
3. To scale across disciplines; for example, from
fluid mechanics to biology (6ersatility).
I shall then conclude the paper with a brief
discussion of three analogies, which are obtained
by applying the FCSA method to these subjects,
and which exhibit the same hyperbolic relation.
1. The analogy between the equations of quan-tum mechanics and those derived using the
FCSA method.
2. The analogies between Mach number (or
Froude number) scaling in fluid mechanics and
the scaling in biology of:
the life span of mammals; and
the incubation period for avian eggs.
1.2. Outline
The paper is divided into eight sections. T
salient characteristics and requirements of rep
cating and non-replicating information syste
are summarized in Section 2. Four examples fro
T-H are used in Section 3 to illustrate and discu
the effects of complexity. Section 4 summari
the key features of the FCSA method. Section
describes the hierarchical levels used in the prese
demonstration of the FCSA method, which
presented in Section 6. The analogies are d
cussed in Section 7, and the paper concludes w
a summary and recommendations in Section 8
2. Replicating and non-replicating systems
Consider the information systems shown in F
1: the salient feature of each is replicatio
whereas the ubiquity of change is the character
tic of its environment. How a system intera
with and responds to these changes determines
evolutionary success or failure.
Replication consists of four information p
cesses or stages: acquisition, storage, retrieval a
transmission. These stages are shown in Fig.
together with their respective requirements a
the means by which the requirements are mThus, the acquisition of information must be
fective, which is accomplished through simplici
The storage of information must be optim
which may be realized by means of parsimo
and synthesis. The retrieval of information m
be fast and easy, both of which are achiev
through efficiency. Finally, the transmission
information must be intelligible, which is arriv
at through the clarity of the message. In biolo
and ecology, the transmission is by means
genes, whereas in the other systems noted in F1, transmission is, according to Dawkins (197
Brodie (1996), Blackmore (1999), via memes.
Information systems may be divided into t
broad groups according to their characterist
and their response to the ever-changing enviro
ment. One group consists of replicating system
the other, non-replicating systems.
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Replicating systems are generally flexible and
adaptable. Thus, they are able to adjust to and
stay current with the changing environment. They
are efficient and, consequently, remain competi-
tive. They survive, replicate, and may be consid-
ered evolutionarily successful.
Non-replicating systems, on the other hand, a
inflexible and maladaptable. They are rigid,
commodating change only with great difficul
Within the context of an ever-changing enviro
ment, they can become obsolete and inefficie
and therefore non-competitive. Eventually, th
Fig. 1. Characteristics of replicating and non-replicating systems.
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systems collapse and disappear. Consequently,
they may be considered evolutionary failures.
I could cite numerous examples of successes
and/or failures for each of the systems shown in
Fig. 1. For purposes of this paper, however, I
shall use only four examples from T-H to illus-
trate how complex information (complexity) in-
evitably leads to failure, in as much as it does not
meet the process requirements noted in Fig. 1.
After discussing the four T-H examples, I shall:
1. demonstrate that the FCSA method provides
the means to meet the requirements of Fig. 1;
and
2. summarize and demonstrate the key features
of the FCSA paradigm for quantifying the
effect (response) that a given agent of change
(environment) has on any of the systems
shown in Fig. 1.
3. Effects of complexity
I have selected four examples to illustrate the
potential effects of complexity on the analysis and
resolution of T-H problems. The first two exam-
ples (the modeling of drop evaporation in mist
flows and of debris dispersal) were chosen to
demonstrate that studies and results, which do not
meet the requirements in Fig. 1, end up on anevolutionary junk heap. In these instances, the
results went unused after being generated and
reported.
The second set of examples (the Reynolds
Stress Equation and multi-fluid formulations, as
currently used in T-H) was selected because they
exhibit the characteristics of non-replicating infor-
mation systems. They do not, therefore, appear to
follow an evolutionary success path.
3.1. Drop e6aporation in mist flows
In the 1970s, the Nuclear Regulatory Commis-
sion (NRC) sponsored an experimental and ana-
lytical research program directed at modeling
evaporation rates of droplets in two-phase mist
flows. This information was needed for a closure
equation in computer codes.
The results of that 2-year effort are shown
Fig. 2. It can be seen that the droplet Nuss
number was correlated in terms of two Reynol
numbers (one for liquid and one for vapor) and
a dimensionless temperature. The correlation w
expressed as a series consisting of 64 terms, w
17-digit accuracy and with coefficients rangi
from 109 to 1036. This is indeed a range of astrnomical proportions. At the time, it prompted m
to comment that were I a droplet, I could n
evaporate, as I would not know how to perfor
such complex calculations!
Looking at these results, one must ask: Wh
information and/or knowledge were acquire
What can be stored? What can be retrieved? Wh
can be transmitted to meet future needs? T
answers to all four questions are identical: no
ing. In the context of Fig. 1, this research effo
produced an evolutionary failure. It not onwasted funds, but, more importantly, wasted
opportunity to impress upon students the value
the efficient production of meaningful and usef
results, as demanded in a competitive technolo
cal environment.
3.2. Debris disposal
Roughly a decade ago, the NRC sponsored
experimental and analytical research program d
signed to model debris dispersal from a reaccavity during severe accidents. The results of th
effort, shown in Fig. 3, were expressed in terms
14 dimensionless parameters and correlated
terms of the functions expressed in Fig. 4.
Observing these results, one might legitimate
ask: What was learned? How was our understan
ing of the process improved? How may we expla
the results? How useful are the results? Again, t
answers to all of these questions are negative. Th
effort also produced an evolutionary failure. T
should come as no surprise, given that the resuing information failed to meet the requirements
Fig. 1. In fact, the results were unintelligible.
3.3. Reynolds stress equation
I shall now summarize some features of t
Reynolds stress equation using information fro
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Fig. 2. A computer-generated correlation for drop evaporation in mist flows.
a highly instructive book by Wilcox (1998). The
summary will serve to illustrate the problems that
lie ahead if trends toward ever-increasing com-
plexity in T-H technology continue unchecked. I
give my views of those problems in the subsequent
section.
The closure problem of Reynolds stress equ
tion is illustrated in Fig. 5, reproduced fro
Wilcoxs book. It can be seen that the averagi
procedure generates six new equations (one f
each component of the Reynolds Stress Tens
and 22 new unknowns. To quote Wilcox:
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This exercise illustrates the closure problem of
turbulence. Because of the nonlinearity of the
NavierStokes equation, as we take higher and
higher moments, we generate additional un-
knowns at each level. At no point will this
procedure balance our unknowns/equations
ledger. On physical grounds, this is not a partic-
ularly surprising situation. After all, such opera-tions are strictly mathematical in nature, and
introduce no additional physical principles.In
essence, Reynolds a6eraging is a brutal simplifica -
tion that loses much of the information contained
in the Na6ierStokes equation. The function
turbulence modeling is to devise approximatio
for the unknown correlations in terms of flo
properties that are known so that a sufficie
number of equations exist. In making su
approximations, we close the system.
I may add that this brute force approach, chaacterized by ever-increasing complexity and d
creasing information content (given t
proliferation of unknowns), does not meet t
requirements of the four information proces
Fig. 3. Proposed similarity parameters.
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Fig. 4. A computer-generated correlation for debris dispersal.
shown in Fig. 1. Indeed, this and similar analyses
should be contrasted to the simplicity, clarity,
informative content and sheer elegance of the
analyses of Kolmogorov (1941a,b), reported more
than half a century ago.
3.4. Multi-fluid formulations
3.4.1. State of the art
Most T-H calculations related to nuclear safety
were performed by computer codes based on the
two-fluid model. Their development was initiated
in the mid-1970s to address the Large Break (LB)
loss of coolant accident (LOCA) issue, and has
been carried on over the past two decades
through several versions of RELAP and TRAC.
The adequacies and shortcomings of these
codes are well established. Supported by a mas-
sive validation process, these codes were used tosuccessfully bring closure to the LB LOCA issue
for conventional NPP. However, time-consuming
and costly modifications were required to address
Small Break (SB) LOCA (in the wake of the TMI
accident), due to the inadequacy of closure equa-
tions (interfacial package), and flow regime
maps, and to difficulties related to numerics and
nodalization. These inadequacies and difficult
were further augmented in the course of applyi
these codes to advanced Nuclear Power Pla
(NPP), Three Mile Island designs. The requis
code modeling improvements were again ti
consuming and costly.
The difficulties in modifying codes to acco
modate new requirements stem, in large pafrom their complexity, i.e. the vast number
closure relations (the constitutive package),
gether with transition criteria and splines, ea
introducing a set of coefficients (dials). The lat
may be adjusted or tuned to produce an accep
able agreement between code calculations and
specific set of experimental data (say, the Pe
Clad Temperature (PCT)). However, this tunin
procedure also generates compensating erro
which limit the applicability of the code to
different set of requirements or design. Ththrough complexity, these codes have already b
come more inflexible and maladaptive to chang
Such complexity, together with the tuning pr
cedure and the exacerbating effect of often inad
quate documentation, render the assessment
code modeling capabilities more a matter of fai
than of reason.
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Such complexity, together with the numerics
and nodalization, also make the two-fluid codes
slow running. For some of the codes, the ratio of
computing time to real time is about 20:1. This
drastically limits the applicability (and thereby the
usefulness) of these codes to advanced NPP de-
signs, in which transients of interest may last for 2
weeks or more.
Two approaches have been used to redress
these shortcomings. One involves three-fluid for-
mulations, while the other applies various averag-
ing techniques to obtain two-fluid equations
containing Reynolds stresses. In my opinion, nei-
ther approach appears to be very promising w
regard to NPP applications, in as much as ea
introduces a new set of unknowns. This, in tu
requires additional closure equations that gen
ate additional coefficients (tuning dials). The d
velopment is open ended.
As is the case with single-phase flow (
Wilcox, 1998), these two approaches only dcrease the information content of a formulati
(by increasing the number of unknowns) witho
any improvement of its physics.
This recourse to ever-increasing complexity
order to reconcile theory with observation
minds me of the increasingly complex calculatio
in astronomy that were required to reconcile t
geocentric concept of the universe with planeta
observations. The complex geocentric model w
accepted as valid (and politically correc
throughout an entire millennium, only to be dcarded (an evolutionary failure) and replaced
the heliocentric model, because of the latte
simplicity, consistency and predictive capabil
(Bronowski, 1973). Is there not a lesson to
learned from such history that would be of bene
to the research and development (R&D) efforts
T-H?
3.4.2. Concerns
In my judgment, one of the greatest concerns
any professional in this field should be the indcriminate use of two- or three-fluid models, whi
invariably claim a good agreement with expe
mental data. Yet, some of these formulations a
codes are known to be inadequate, flawed and/
incorrect. The good agreement may be explain
only in terms of the carefully tuned dials hidd
in the code, as noted by Travkin and Catt
(2000).
Although good agreement with experimen
data may ensure the continuation of project fun
ing, such formulations cannot contribute to tfund of knowledge. Laws of variable coefficien
and tuning dials are not yet laws of physics.
Yet these continuous claims to success ha
institutionalized the art of tuning as an accep
able methodology for addressing and resolvi
technical issues and/or scientific problems. I cou
apply to this modern art of tuning, perhaps tFig. 5. Closure problem with the Reynolds stress equation
(Wilcox, 1998).
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Fig. 6. Success path.
notion of tuning spins, in an analogy to the
acknowledged spins used to address and resolve
political issues. It would thus appear that spin
doctors are not limited to political circles!
Such comments should not be construed as
opposition on my part to the use of codes such as
RELAP and TRAC; this is not at all the case. These
codes are adequate for the purpose for which they
were originally designed, extensively tested and
validated. My concerns relate to the use andmodification of these codes to accommodate new
designs and/or meet new requirements (for speed,
repetitive calculations, long-lasting transients, etc.)
brought about by the de-regulation of the power
industry (a changing environment). The increasing
complexity inherent in such an approach leads
inevitably to the evolutionary junk heap.
What is needed now, and will be all the more vi
in the future, are flexible, accurate and efficie
codes to maintain a competitive edge in a changi
environment. I cannot emphasize here enough th
speed and accuracy are not incompatible requi
ments. They may both be realized through mod
larity and flexible architecture.
For these codes to be evolutionary success
they must have the characteristics of the replicati
information systems illustrated in Fig. 1. Goingstep further, the stages and requirements of
success path for such code development efforts a
presented in Fig. 6, which reflects and is found
upon the lessons I have learned through lo
involvement with this technology.
The steps and requirements are so obvious, th
one might wonder why Fig. 6 is included in th
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paper. The answer is quite simple. Just 1 year ago,
I attended a review meeting concerned with a new
code development program. To my amazement,
not only were the elements of the program not
integrated, but fewer than one-third of the steps
in Fig. 6 had even been considered! As a result of
that meeting, I felt compelled to prepare Fig. 6
and attach it to my evaluation memorandum. I
am including it in this paper with the thought that
it might perhaps be of some use to a broader
audience.
4. FCSA method
I would hope that the four examples in the
preceding section suffice to illustrate the effects
that increasing complexity has (or inevitably willhave) on T-H analyses.
In contrast to complexity, simplicity has long
been recognized as the desideratum for any sci-
ence. Let me quote a few such pronouncements.
Ockham: Pluritas non est ponenda sine necessi-
tate; paraphrased variously by Jeffrey and
Berger (1992), It is vain to do with more what
can be done with less. An explanation of the
facts should not be more complicated than
necessary. Among competing hypotheses, fa-
vor the simple.Gibbs: One of the principal objects of practi-
cal research is to find the point of view from
which the subject appears in its greatest
simplicity.
Maxwell: The greatest desideratum for any
science is its reduction to the smallest number
of dominating principles.
In what follows, I shall demonstrate that these
requirements (simplicity, parsimony and synthe-
sis) are the key features of the FCSA method.
Indeed, it was statements such as those quotedthat guided the development of this paradigm,
which quantifies the effects of a changing environ-
ment upon an information system. For reasons
cited in Section 1.1, these same features enable the
FCSA method to process information with effi-
ciency and versatility through the stages shown in
Fig. 1.
In these times of rapid change vis-a-vis techn
ogy and economy (engendered by globalizatio
and the age of information), efficiency and v
satility are of utmost importance, not only
industrial, but also to educational systems. T
days graduates must be flexible, adaptable a
versatile in their professional careers, lest th
become obsolete and, perhaps, unemployable.
The FCSA method originated during the cou
of a program designed to scale severe acciden
(Zuber, 1991). A summary of that effort w
recently published by Zuber et al. (1998). Af
1991, developments were reported by Zub
(1993, 1994, 1995). In this paper, I shall outli
the method and summarize some of the recen
published results (Zuber, 1999), omitting much
the commentary and detail, to illustrate its pote
tial benefits to T-H.To emphasize the generality of the method
ogy, I shall continue to employ words and expr
sions (such as information system, agent
change, cell, influence length, signal, etc.) that a
applicable to many different disciplines. I do th
deliberately, as words are often codes for a part
ular message, and therefore tend to restrict t
generality and import of a concept.
However, for purposes of this paper relating
T-H, the information entities of interest are ma
momentum and energy, and the agents of chan
are fluxes of mass, momentum and energy acro
system boundaries. In this paper, therefore, t
application and demonstration of the FCS
method will be carried out in terms of these thr
variables and parameters.
4.1. Spatial and temporal scales
Consider a signal being transferred across
area, A, and being felt (integrated) within a vume, V, which will be referred to as the inform
tion containing volume or receiver.
As the volume is a measure of the capacitanc
and the transfer area gauges the intensity of
process, the spatial scale that characterizes
transfer process is the transfer area concentratio
defined by
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1
u=
A
V(1)
which henceforth will be referred to as the influ-
ence or characteristic length u.
One can look at boundary conditions as agents
of change or constraints imposed upon the trans-
fer area that induce changes within the informa-tion containing volume. Similarly, one can look
upon these changes as the response of the en-
closed entity (amount of information) to the con-
straint-induced perturbations at the boundary.
Therefore, the rate of change of such an entity is
determined by the time constant for this internal
accommodation process. We shall consider sepa-
rately the effects of constraint and of the
response.
We define M as the metric for the entity con-
tained in V, and b as the agent of change.Clearly, both M and b are problem specific. For
applications discussed in this paper, M will denote
momentum or energy, whereas b will represent
forces, fluxes or power. The rate of change of M
due to the action of b is then
dM
dt=F (2)
To obtain the time constant for the receiver, we
define as the fractional rate of change (FRC) of
M. Thus,
=1
M
dM
dt(3)
In as much as we consider perturbations from a
steady state, M0, it follows from Eqs. (1) and (2)
that the FRC can be expressed as:
=F
M0(4)
which implies a linear approximation of Eq. (3).This approach has four important features.
First, the kinetic aspect of the problem is ac-
counted for by Eq. (2).
Second, the kinematic aspects are reflected in
Eq. (3), in that the FRC introduces, via the
concept of action (discussed in Section 4.3), the
kinematic relation that specifies the process signal.
Therefore, the FRC provides the temporal sc
for a particular transfer process in the receiver
Third, the kinematic and kinetic aspects a
combined, via the effect metric V (see Section 4
to produce the scaling criterion for the trans
process of interest.
Fourth, it is this decomposition of the proble
into kinetics and kinematics that allows the usea single method to analyze and scale differe
transfer processes.
The second temporal scale depends upon t
time, ~, during which the change is being o
served, i.e. integrated. We shall denote it as clo
time to differentiate it from the process time sc
1/.
For an open, flow system, the clock time
defined by
1~=Q
V
where Q is the volumetric flow rate. It can be se
that ~ is a function of the spatial scale an
therefore, of the hierarchical level at which t
change is to be observed. For one-dimension
flows through ducts with constant cross-section
area, ~ becomes a function of length and of t
average velocity, 6, of the fluid.
In this paper, we consider transfer processes
three hierarchical levels, macro, meso and micrwhich are identified by three spatial scales: Lsthe system, u of the cell and uk of the dissipatio
We therefore shall use three scales for the clo
time ~. Thus, for the system:
~s=Ls
v
for the cell:
~c=
u
vand for dissipation:
~k=uk
6k
where vk is the dissipation velocity discussed
Section 6.3.
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4.2. The effect metric d
Having defined the FRC of M by , and the
clock time by ~, we now designate the metric for
change as
V=~ (9)
Since ~ is the time for observing (integrating) achange, it follows from Eqs. (2) and (4) that
V=F
M0~=
MM0
M0(10)
Consequently, while the FRC is the metric
for the intensity of a transfer process, the metric V
denotes the fractional change (%) of the informa-
tion metric M during a period ~, as a consequence
of the information transfer rate b. For this rea-
son, we shall refer to V as the effect metric (the
effect ofb on M during ~).Consider now processes for which the effect
metric V is a constant, say V0. It follows then,
from Eq. (9), that
~=V0 (11)
which plots a hyperbola as shown in Fig. 7. We
note that this hyperbola divides the ~ plane
into two regions.
In region (1), the hyperbola V0 limits the clock
time ~, during which a given FRC (call it 1) can
be sustained. As characterizes a particulartransfer process b1, the hyperbola V0 delineates
the region within which this process can be main-
tained. In region (2), the metric M can no longer
sustain the fractional change at the rate 1.
Therefore, the process either ceases or must
change to a completely different kind of process.
Section 7 will demonstrate the following.
1. In compressible flow, the effect metric V is t
inverse of the Mach number. Therefore, t
hyperbola V0=1 divides the ~ plane in
two regions: the subsonic (V\1) and the s
personic (VB1).
2. The life span of mammals scales according
such a hyperbola, so that, indeed, small mammals (e.g. mice) have high metabolic rates a
short life spans, while the opposite holds tr
for large mammals (elephants), which have
low , but a long residence time, ~. Thus, f
all mammals, the hyperbola V0 separates t
region of life (region (1)) from that of t
beyond (region (2)).
3. The time to hatch eggs also scales as a hyp
bola. Thus, for avians, the hyperbola V0 sep
rates the egg time (region (1)), from t
flying time (region (2)).
4.3. Kinematic and kinetic aspects
We proceed by defining three importa
parameters.
Process velocity: 6p=u (
Process action: Ap=6pu=u2 (
Flow action Af=6u (
Given that the FRC is the temporal scale fthe intensity of a particular transfer process, th
6p is the metric for its speed.
The rationale for using the concept of action
a parameter is discussed elsewhere (Zuber, 199
Briefly, it is a parameter that characterizes qua
tum, diffusion and wave phenomena. In t
present application, it provides the key to unco
pling kinematics from kinetics.
The action parameters Ap and Af are intr
duced in the formulation via the effect metric
Considering a cell and its clock time, ~, it follofrom Eqs. (7), (9), (13) and (14) that the eff
metric V can be expressed as:
V=u
6=
Ap
Af(
which clearly illustrates the kinematic features
V. However, by means of Eq. (10), we can alFig. 7. The plane and effect metric V0 (Zuber, 1994, 1999).
Note: high frequency, short life; low frequency, long life.
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Fig. 8. The road map to the effect metric and four similarities.
the fractional rate of change (FRC) (step
in Fig. 8), which specifies the temporal scale f
the change;
the cell clock time ~c, which specifies the du
tion of the change within the cell (step 2);
the effect metric V, which quantifies the eff
(fractional change) that b had on M duri
clock time ~c (step 3); and two action parameters, Ap and Af, the first
which (step 4) specifies the process (a
thereby the signal), and the second that
counts for the flow (step 5).
By means of the two action parameters, t
effect metric V may be expressed in terms of t
kinetic and kinematic aspects of the process (st
6). The first relation accounts for the agent
change (the environment), and the second for t
speed of the signal within the cell (the speed
accommodation, of adjustment).The relations in Fig. 8 (steps 3 and 6) demo
strate that a transformation (scaling) that p
serves the effect metric V automatically a
simultaneously satisfies and ensures four simila
ties: kinematic, kinetic, temporal and fractiona
The concepts of a cell (a quantum of volum
defined by the influence length u, and of t
fractional rate of change , impart simplicity a
generality to the FCSA method. The concept
two action parameters, Ap and Af, introdu
synthesis and parsimony. When these concepts acombined and expressed as the effect metric
the FCSA provides a general, efficient, flexib
and adaptable method for quantifying changes
information systems induced by an agent
change.
In this lecture, these attributes will be demo
strated by applying FCSA to momentum and
energy at three hierarchical levels.
5. Hierarchical levels
Given that I intend to demonstrate the app
cability of the FCSA method to processes a
systems in which the spatial scale varies over
or more orders of magnitude, I shall structure t
demonstration and interpret the results within t
context of the theory of hierarchical systems. T
express V in terms of the kinetic parameter b.
Thus, we can write:
Ap
Af=V=
F
M0
u
6(16)
thereby demonstrating the dual interpretation of
V as both kinematic and kinetic.
4.4. Features and road map
The road map for applying the FCSA method
to an information system is displayed in Fig. 8,
which also outlines its key features.
FCSA considers an entity M, contained in a
volume V, acted upon by an agent b, which
induces changes in M. The rate of change of M is
specified by the signal that propagates within the
volume V. The latter depends upon the transfermode, i.e. whether it is by waves, diffusion, vortic-
ity or a combination thereof.
Information containing volume V, defined by
the influence length u, is referred to as a cell in the
hierarchical scales considered in this paper. At
that level, FCSA considers five parameters (con-
cepts), which include:
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relevance of this theory was recognized and used
to develop a hierarchical approach to scaling (Zu-
ber, 1991; Zuber, 1998) and was adopted in the
studies of Reys and Hochreiter (1998), Ishii et al.
(1998), Peterson et al. (1998).
One of the key features of hierarchical struc-
tures is that the question (information needed)
determines the hierarchical level at which the an-
swer is to be found. Thus, lower levels provide
more detailed information than those higher up.
The three levels shown in Fig. 9 reflect the follow-
ing three posed questions, and illustrate an appli-
cation of the FCSA method.
Taking momentum as the subject of interest, we
start from the top, i.e. from the macro level, and
address the overall problem with the initial
question:
1. What parameters scale the change of momen-tum of the entire system?
2. Having identified the overall parameters at the
macro level, we proceed to the meso level with
a more detailed question:
3. What processes affect the overall parameters,
and how are these processes scaled?Having
identified the processes and the relevant scal-
ing relations at the meso level, we proceed to
the micro level, with a still more detailed
question:
4. How are the changes of momentum tra
formed in an irreversible loss, and what sca
the dissipation rate?
The implementation of this procedure is a
shown in Fig. 9.
At the macro level, the system specifies t
spatial scales that are used in Eq. (4) to determi
the FRC of momentum m,s for the system. Tsystem also determines the clock time ~s. Th
two parameters generate the effect metric for t
whole system, V, which both scales the change
momentum and identifies the overall governi
parameters.
The meso level is defined by the influence leng
u, which specifies the dimensions of a cell with
which the process is to be addressed in mo
detail. A particular process is specified by t
process action Ap, which is used to obtain t
FRC of momentum m,p for that process. The calso determines the clock time ~c. These t
parameters (m,p and ~c) generate, in turn, t
effect metric Vp, which scales that particu
process.
At the micro level, the assumption of isotrop
ity specifies the fluid action Af, which, togeth
with the FRC m,p, generates all parameters th
characterize the dissipative level (i.e. the FRC
momentum k, the spatial scale uk, the velocity
and the clock time ~k). These parameters in tu
generate the effect metric Vk, which turns outbe equal to unity, reflecting a compl
dissipation.
In the following section, it will be demonstrat
and confirmed that the same method (FCSA) a
the same metric (V) may be applied to the hiera
chical levels shown Fig. 9, and may be used
generate of the appropriate scaling criteria
reflect the degree of interest and details of ea
level.
6. Demonstration through applications
6.1. Macro-le6el scaling
We consider four examples, which are ill
trated in Fig. 10. The first deals with scali
pressure drop in flows through ducts. The metFig. 9. Fractional change scaling at three hierarchical levels
(Zuber, 1999).
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Fig. 10. Examples of sealing at the macro/system level (Zuber, 1994, 1999).
M is therefore the momentum (zV6), and the
agent of change b is the stress force at the wall
(|wA). Inserting these (system) parameters in Eq.(4) yields the FRC of system momentum m,s(expressed in terms of the friction factor f ) that,
with the system clock time ~s, generates the system
momentum effect metric Vm,s. It can be seen that
it is identical with the definition of the friction
loss factors given in Bird et al. (1960).
The second subject is the scaling of heat trans-
fer by forced convection. The metric M is now the
enthalpy (zVcpDT), and the agent of change b is
the heat transfer rate from the wall (hAD
T, whereh is the heat transfer coefficient). Following the
same procedure as already described, we obtain
the FRC of system enthalpy, e,s, and the system
enthalpy effect metric Ve,s, which is now a func-
tion of the Stanton number, St, and identical to
relations given in any standard textbook on the
subject.
The third topic deals with the scaling of a loss
of coolant accident (LOCA) in nuclear reactors.
The metric M is now the enthalpy (zVcpDT), in
the primary side, whereas the agent of change b isthe reactor power. Thus, we obtain the FRC e,sof the enthalpy in the reactor, and the effect
metric Ve,s. The system clock time ~s is given by
Eq. (5), where Q is the volumetric flow rate of the
fluid through the break. If fluid properties and
clock time are preserved, then the effect metric
Ve,s becomes the power to volume scaling rule:
F1
V1=F2
V2(
This scaling rule was used to design and operatest facilities that produced experimental data ne
essary to validate large computer codes, which,
turn, were used to address and resolve saf
issues relating to LOCA (Boyack et al., 199
Zuber et al., 1990).
The final topic deals with the life span of ma
mals. It is included here only to demonstrate th
the FCSA method may be applied to situatio
for which differential equations are not availab
and the processes are not sufficiently well und
stood to use the Buckingham ^ Theorem. I shreturn to this subject in Section 7.
We note that the effect metrics V scale pressu
drop and heat transfer at the system level, a
identify the overall parameters that must be co
sidered. However, they provide no information
to what processes affect these two paramete
This information must be sought at the me
level.
The effects of two or more agents acting sim
taneously on a system are discussed in Append
A, in order to illustrate the efficiency that may realized through scaling.
6.2. Meso-le6el scaling
At this level, we analyze processes that occ
within the volume of a cell defined by the infl
ence length u. Within such a cell, we consid
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and 3) to obtain the effect metric for the momen-
tum Vm (step 4.)
We use the kinematic branch ofVm to identify
three types of flows: those for which (1) Ap is
constant, (2) Vm is constant, and (3) neither Apnor Vm is constant. These three types characterize
three transport processes.
For diffusion-dominated processes, the transferis by molecular action. Consequently, the process
action Ap is constant and equal to twice the
kinematic viscosity.
For vorticity-dominated (high-velocity) flows,
the transfer is independent of molecular effects.
The process action Ap must therefore be propor-
tional to Af, with Vm as a constant, say V0. Thus,
Ap=V0Af (22)
By introducing the definitions of the actionparameters (Eqs. (13) and (14)), we can express
Eq. (22) as
mu
6=V0 (23)
indicating a constant vorticity flow.
For flows affected by both diffusion and vortic-
ity, we interpolate between these two limits by
defining an eddy action Ae in terms of a general-
ized geometric mean
Ae=Ad1mAv
m (24)
which reduces to flows dominated by diffusion for
m=0 and by vorticity for m=1.
Expressing the kinematic branch ofVm in terms
of three process actions (step 5) leads to three
effect metrics Vm (steps 6, 7 and 8), i.e. one for
each mode of transfer.
Referring to the results shown in Fig. 11, we
observe the following:(1) For flows through circu-
lar ducts (with a diameter D), the influence length
(defined by Eq. (1)) is
u=D
4(25)
and the cell Reynolds number Reu becomes:
Reu=6u
26=6D
86=
Re
8(26)
(2) For diffusion-dominated flows through circ
lar pipes (step 6), the friction factor f becomes
f=16
Re(2
which is identical to the relation derived by H
genPoisseulle for viscous flows. Note, howev
that Eq. (27) was derived here without even wring the differential equation, much less solvi
it.(3) For vorticity-dominated flows over a rou
surface, the friction factor f depends only up
the geometry and configuration of the roughne
As
u
6=~c=V0 (2
these flows plot as hyperbolae in the ~ pla
shown in Fig. 2.(4) By varying the exponent m
Eq. (24), we obtain different relations between tReynolds number and the friction factor in step
Thus, when m=0.75, we obtain fReu-0.25, whi
is the relation proposed by Blasius for turbule
flow. Whereas, when we take m=0.80, we obta
fReu-0.20, which is the relation proposed by H
mann, also for turbulent flow.(5) The kine
branch ofVm (step 4) is an energy ratio, which
expressed in terms of the friction factor in steps
7 and 8. This branch does not specify (accou
for) the type of transfer. The kinematic side do
that. This observation should come as no surpriin as much as flow pattern is a metaphor f
kinematics (and vice versa).
6.2.1.2. Wa6es and 6ibrations. I shall now outli
the application of the FCSA method to wave a
vibrations, summarizing the findings of Zub
(1999).
Once again, we identify the metric M w
momentum (Vz6) and the agent with the p
turbation force lF, at the boundary, to obtain t
FRC m. Thus,
m=lF
Vz6(2
For a particular problem of interest, we rela
the perturbing force lF to the specific energy
the displacement lu, and express the FRC mterms of these two parameters. Thus,
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Fig. 12. The effect metric V for various types of waves and vibrating systems (Zuber, 1999).
m=E
6ulu
u(30)
Following the procedure illustrated in Fig. 8,
we introduce the action parameters Ap and Af to
express the effect metric in terms of the kinematic
and kinetic branches:
Ap
Af=Vm=
E
62
lu
u
(31)
Again, we use the kinematic branch to specify
the process. Consequently, for waves, we identify
the process action Ap with the wave action defined
by
A=E
(32)
In discussing wave phenomena, Witham (1974),
Johnson (1997) note that the wave action, defined
by Eq. (32), is a quantity even more fundamental
than the energy, in as much as it is conserved,while neither wave energy or frequency is.
Substituting Eq. (32) into Eq. (31) yields:
1
m=
lu
6(33)
which, upon substitution into Eq. (30), generates
the FRC m for waves and vibrations:
m=Eu
(3
whereupon the effect metric for the cell becom
Vm=m~c=E6
(3
Fig. 12 shows the results obtained by applyi
the FCSA method to various types of wave ph
nomena and vibrating systems. Additional exaples are not included, given the limitations on t
length of this paper. This figure illustrates t
following.
For pressure waves (where c is the velocity
sound), Vm is identical to the inverse of t
Mach number.
For surface waves in deep water (where u is t
wavelength), Vm is identical to the inverse
the Froude number.
For long waves (where p0 is the depth of t
channel), Vm is again identical to the inversethe Froude number.
For a vibrating string (where E is the longitu
nal elasticity), the FRC m becomes the f
quency of vibrations.
For a spring-mass cell (where k is the spri
constant, u its length and m the particle mas
the FRC m is the frequency of oscillations
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For an array of n spring-mass cells, the FRC
m is again the frequency of oscillations.
I trust that these applications were sufficient to
demonstrate that FCSA provides a single method-
ology to scale and to analyze transfer processes
associated with particles, waves, diffusion and
vorticity.
6.2.2. Fractional change of energy due to
dissipation
In this section, I shall outline the application of
FCSA to scale and analyze the effect of dissipa-
tion on the kinetic energy, summarizing the re-
sults of Zuber (1999).
For this application, the metric M is the kinetic
energy (zV62/2) and the agent of change b is
the rate of dissipation due to the stress force
(F|=|A.) We let the volume V be in contact with
the wall, and evaluate the stress there to obtainthe FRC of kinetic energy kE. Thus:
kE=|wA6
zV(62/2)=2
E
6u=2m (36)
The effect metric VkE may be expressed, there-
fore, in terms ofm or Vm to yield the well-estab-
lished scaling criterion
VkE=kE~c=2m~c=2Vm=f (37)
and be evaluated by means of the three relations
derived in the preceding section (steps 6, 7 and 8in Fig. 11.)
We proceed to scale and analyze the rate of
dissipation by means of the FCSA method. To
this end, we define the specific dissipation rate by:
m=F|6zV
(38)
and express it in terms of the FRC kE from Eq.
(36). Thus:
m=kE62
2=m62 (39)
We use this relation in the road map (see Fig.
13) that shows two parallel paths: one for m, the
other for m. The parallel paths serve two purposes.
They reveal the relationship between the specific
energy E and the specific dissipation rate m, as well
as demonstrate how each relates to the other
parameters.
Following the FCSA method, we introduce t
action parameters Ap and Af, and obtain t
invariant (step 4) and the specific dissipation ra
m in terms of three parameters (step 5.)
Again, we use the process action Ap to spec
the transfer processes: one for diffusion-dom
nated and the other for vorticity-dominated flo
(step 6). The third mode (for combined transfshown in Fig. 11 has been omitted for the sake
simplicity. These two expressions for Ap lead
two relations for m (steps 7 and 8) and t
relations for m (steps 9 and 10), each valid for t
specified transfer process.
The relations for vorticity (step 7) and for
diffusion-dominated process (step 8) are identic
to the relations shown in Fig. 11 (steps 7 and
respectively), albeit expressed in terms of differe
parameters.
The dissipation rate for diffusion-dominatflows (step 9) can be expressed in terms of t
pipe diameter D and average velocity 6. Th
from Eqs. (14) and (25), we have
m=646
D262
2(4
valid for HagenPoisseulles flow.
The dissipation rate for vorticity-dominat
flows (step 10) is the Kolmogorov-5/3 equati
for the inertial subrange (Kolmogorov, 1941a,To demonstrate this statement, we shall expr
the dissipation rate given by
m=1
V01/2
E3/2
u(4
in terms of the parameters used by Kolmogoro
i.e. in terms of the wavenumber k.
k=1
u(4
and of the energy spectrum E(k):
E(k)=E
k=Eu (4
We can therefore express Eq. (41) in terms
these parameters to obtain:
E(k)=V01/3m2/3k5/3 (4
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which is Kolmogorovs equation when the con-
stant V01/3 is identified with Kolmogorovs con-
stant Ck.
We note, in closing, that Eq. (44) can also be
derived (Zuber, 1999) by considering a wave
eddy duality in analogy to the wave-particle dual-
ity in quantum mechanics. The only difference
between these two approaches is the exponent of
the constant V0. For the waveeddy duality, the
exponent is 4/3 instead of 1/3 in Eq. (44). The
approach described is preferable, in as much as no
waveeddy duality is invoked in the derivation of
Eq. (44).
6.3. Micro-le6el scaling
In this section, we shall apply the FCSmethod to scale and analyze the dissipation raat the micro scale, summarizing the results Zuber (1999).
As the spatial scale decreases, diffusion b
comes the dominant transfer process (for a discusion, see Zuber, 1999. Consequently, we start wthe relations shown in steps 8 and 9 in Fig. 13
Consider next the effect of isotropy. Fisotropic conditions, the two action parametmust be equal. Consequently, we identify Af w2w, thereby reflecting a random motion (step 1
Fig. 13. Road map for scaling the specific dissipation rate at the meso and micro levels.
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We then introduce the relations from step 11
into those of steps 8 and 9, which leads to equa-
tions that are valid at the micro level (steps 12, 13
and 14). These equations merit further comment.
First, the three identities in step 13 are identical
(except for the numerical factor of 2) to the
relations derived by Kolmogorov, and which
define the Kolmogorov scale for dissipation.Consequently, we have denoted them by the
subscript k. In transforming one identity to an-
other, we made use of Eqs. (13) and (14).
Thus:
Ap
2w=kuk
2
2w=
6kuk
2w=1 (45)
The last identity corresponds to a Reynolds num-
ber for dissipation: Rek=2.
Second, given that the specific energy E is deter-
mined by the constraint, the three equations insteps 12 and 14 show that the spatial length uk,
velocity 6k, and the specific dissipation rate m are
determined by the constraint.
Third, comparing the third equation in step 13,
where m is expressed in terms of Kolmogorovs
dissipation velocity 6k, with Eq. (19), for the
friction velocity 6, it can be seen that these two
equations are identical. Consequently, these two
velocities correspond to each other, and
6
2
=6k2
=E (46)which, for a given specific energy E, is the
parameter for scaling velocities.
Fourth, the effect metric Vk at the micro-scale
level is expressed in terms of FRC k and the
micro-level time clock defined by Eq. (8). Thus,
Vk=k~k=kuk
6k
(47)
which, in view of Eq. (45), becomes
Vk=1 (48)indicating that, at the micro level, the entire en-
ergy has been dissipated during the microlevel
clock time ~k.
I hope that the applications presented have
both demonstrated and confirmed that the FCSA
method provides a single paradigm for scaling and
analyzing transfer processes across hierarchical
scales ranging from nuclear reactors to K
mogorovs micro scale.
7. Analogies
In this section, I would like to briefly note thr
analogies: one between the concepts and equtions summarized in the paper and those in qua
tum mechanics; and the other two between flu
dynamics and two biological processes. Giv
that these analogies deal with topics outside t
field of T-H, I shall not discuss them in any det
(these subjects will be covered in separate public
tions.) I am citing them here merely to illustra
the versatility of the FCSA method.
7.1. Analogy with quantum mechanics
We have already noted that Eq. (21) is t
mathematical analog of de Broglies equation
quantum mechanics. Fig. 14 shows that this an
ogy may be extended to other parameters a
equations.
The analogies illustrated now are mathematic
in nature. However, there is also a conceptu
analogy to be drawn, with potentially significa
implications.
Consider de Broglies equation. It relates to t
specific energy E and to two velocities, one for twave and the other for the particle. The fi
characterizes the kinematics of the process, wh
the second characterizes the kinetics. Thus,
Broglies equation displays and combines bo
features (kinematic and kinetic) of the problem
Consider now Eq. (21), which is based upon t
concept of fractional change and was derived
uncoupling the kinematics from the kinetics.
also has two velocities: one for the process (
and the other for the fluid particle (6).
By introducing two action parameters, one fprocess (Ap) and the other for flow (Af,) a
relating them to particular flow processes,
were able to identify a process velocity (6p) f
each flow pattern. As a result, Eq. (21) was us
to scale and analyze not only wave processes (
does de Broglies equation), but other processes
well, including diffusion, vorticity and their co
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Fig. 14. Analogies (Zuber, 1999).
bined effect, dissipation, single-particle and multi-
particle vibrations, etc.
Therefore, by developing and utilizing the con-
cept of equal fractional change in conjunction
with the concept of action, we have extended the
relevance and usefulness of de Broglies equation
to diverse processes and numerous applications.
7.2. Analogies with biological processes
I shall now comment briefly on the analogy
between fluid dynamics and two biological pro-
cesses; specifically, the life span of mammals and
the hatching time of avian eggs.
Biological studies of mammals have shown that
an allometric relation correlates the observed
metabolic rates as a function of body weight, with
an exponent of approximately 0.25, whereas
the life span correlates with an exponent of ap-proximately +0.25 (Kleiber, 1932; Stahl, 1967;
McMahon, 1984, and others.) Therefore, their
products yield the hyperbolic relation shown in
Figs. 2 and 15a.
Thus, for mammals, the hyperbola V0 is a
dividing line that separates the region of life (re-
gion (1)) from that of the beyond (region (2)).
In a similar manner, one can obtain a hype
bolic relation (Fig. 15a) for the time to hat
avian eggs. In this case, Region (1) corresponds
the egg time, whereas region (2) is the flyi
time for avians.
The hyperbola V0 in Fig. 15a indicates th
time has the same (symmetric) effect on the l
cycle of each species. In as much as life is specifi
by a hyperbola, different stages are specified
hyperbolae VBV0 . For example, species rea
their reproductive stage at V=0.25V0, matur
at V=0.50V0, lose their reproductive capacity
V=0.75V0 and die at V=V0. The effect of tim
is the same, i.e. life is time symmetric.
The effect metric V in Fig. 15a translates t
energetic relation of a process into a tempo
relation. The constancy ofV0 implies time symm
try and conserves the energy ratio. This illustrathe theorem of Noether (1918) (Mills, 199
Oliver, 1994, and others), which states that
every symmetry in nature, there corresponds
conservation law. Thus, the energy conservati
law corresponds to time symmetry. We may the
fore conclude that life is a manifestation
Noethers theorem.
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We have shown in Section 6.2 that, for com-
pressible flow, the effect metric Vm is the inverse
of the Mach number. Consequently, lines of con-
stant Vm plot as hyperbolae in the ~ plane
(Fig. 15a,) and as straight lines in the 66pplane (Fig. 15c.) Thus, the metric Vm=1 sepa-
rates the subsonic flow (Vm\1, region (2)) from
the supersonic flow (VmB1, region (2)). In this
application, the effect metric translates the kinetic
relation of the process into a kinematic one.
These observations indicate that the process of
compressible flow and mammal life share several
analogous features.
There is an additional interesting observation
that may be made with respect to this analogy. It
was observed by von Karman that a shock wave
divides space into two regions. Ahead of t
shock is the region of silence, of no informati
(see Fig. 15c,) since the velocity of the sign
(announcing the change) is lower than that of t
particle (airplane). Information is available b
hind the shock wave, given the awareness of
passage.
While von Karman considered supersonic flo
in the context of space and information, the an
ogy shown in Fig. 15 suggests that life may
considered in the context of time and informatio
Thus, as we move through time (or as time pas
us by), we have no information as to what
ahead, or of what the future holds. Information
available to us only from the past (or what
behind us in time).
Fig. 15. Analogy between the life span of mammals and fluid dynamics.
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8. Summary and recommendation
This lecture had two objectives: to contrast the
effects that complexity and simplicity have (and
will continue to have) on R&D efforts in T-H;
and the other, to demonstrate that scaling pro-
vides the means by which to process information
in an efficient manner.The complexity versus simplicity comparison
was made in the context of replicating and non-
replicating information systems, which allowed
the following.
1. To illustrate how ever-increasing complexity in
formulating and analyzing T-H problems in-
evitably leads to inefficiency, obsolescence and
evolutionary failure.
2. To note that simplicity, which allows for parsi-
mony, synthesis and clarity of information,
ensures efficiency, survival and evolutionarysuccess.
3. To identify the requirements (and the means
of achieving them) that a successful R&D
effort must have.
4. To provide a success path for an R&D effort
to follow.
To meet the second objective, I summarized the
key features of the Fractional Change, Scaling
and Analysis method (FCSA), which are simplic-
ity, parsimony, synthesis, efficiency and versatil-
ity. These features were demonstrated by applyingthe FCSA paradigm to various processes. It confi-
rmed that a single concept (simplicity and parsi-
mony) and a single methodology (again, simplicity
and parsimony) may be used to:
1. scale all transfer processes associated with par-
ticles, waves, diffusion and vorticity (synthesis)
across hierarchical levels ranging from Kol-
mogorovs micro scale to a nuclear reactor
(synthesis and efficiency);
2. derive Kolmogorovs scaling relations for:
The inertial subrange and the micro range(synthesis); and
3. scale across disciplines; for example, from fluid
mechanics to biology (6ersatility).
In the context of the fluid dynamics-life span
analogy of Fig. 15, I cannot know what lies
ahead. However, I can integrate information from
the past to conclude that, by pursuing the path of
ever-increasing complexity, the T-H technolo
will follow doggedly in the footsteps of the do
bird. I sincerely hope that this will not be the ca
Appendix A. Efficiency through scaling
This appendix has a twofold purpose: first, illustrate the inefficiency and, therefore, the was
fulness, associated with computer code safe
analyses as presently conducted; and, second,
demonstrate, through a simple example, the sa
ings (in terms of effort, time and funds) that m
be realized through scaling, and the increas
efficiency thereby afforded a safety analy
process.
To my knowledge, the results of LB and
LOCA safety studies (either experimental or co
puter based have never been cast in a dimensioless form by means of appropriate scali
relations. Thus, a full synthesis of the resu
(information in the context of Fig. 1)) has n
been achieved.
The consequences of this lack of synthesis we
(and still are) inefficiency and wastefulness, in
much as each power level, each break size, ea
break location, each reflood rate, etc., require
separate experiment and a separate computer c
culation, both of which are time consuming a
expensive.The already onerous task of having to consid
separately the effects of varied and numero
parameters (a piecemeal approach, at best) w
greatly augmented by the de-regulation of t
power industry, which introduced competition
the environment. The attendant quest for e
ciency generated, in turn, the need for best es
mate (BE) calculations and for quantification
uncertainties, which, to be done properly, requ
numerous sensitivity calculations.
Although scaling provides the technical ratnale and methodology for reducing the number
parameters in an equation (by casting the equ
tion in a non-dimensional form and expressing
in terms of scaling groups), this, as already note
was not performed in conjunction with BE calc
lations. Instead, arm-waving arguments are us
to justify a reduction in the number of sensitiv
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N. Zuber /Nuclear Engineering and Design 204 (2001) 1 27
calculations, for the sole purpose of reducing,
thereby, the cost of a safety analysis.
It is my considered opinion that such safety
analyses are bound to be most detrimental to
nuclear power technology. There are many lessons
that should have been learned from past experi-
ence and history. However, as my views on this
subject have been expressed in various documentsavailable to the public, I shall not debate the issue
further here. Instead, I shall endeavor to illus-
trate, by means of a simple example, the savings
and efficiency that may be realized through scal-
ing, and to contrast these positive effects to the
wastefulness and inefficiency of a piecemeal
approach.
Consider a fluid of density z, specific heat cp,
flowing at a mass flow rateW, through a vessel of
volume V, heated by two sources. For a well-
stirred vessel, it may be assumed that the fluidtemperature in the vessel is equal to that at the
outlet. Consequently, the energy balance becomes:
VzcpdT
dt=Wcp(TinT)+h1A1(T1T)
+h2A2(T2T) (A1)
where Tin is the temperature of the fluid at the
inlet, T1 and T2 are the temperatures of the first
and second sources, respectively, and h and A are
the heat transfer coefficient and transfer area of asource.
In the context of the FCSA method, the metric
M is the enthalpy in the vessel, whereas the agents
of change are the two energy sources, each of
which generates a FRC of enthalpy , thus:
h1A1
zcpV=
h1
zcp
1
u1=1 (A2)
h2A2
zcpV=
h2
zcp
1
u2=2 (A3)
By means of these two parameters and the
definition of the system (vessel) clock time given
by
1
~s=W
zV(A4)
we can transform Eq. (A1) into
~sdT
dt=TinT+V1(T1T)+V2(T2T) (A
Thus, each source (agent) has its own effect met
V.
Inas much as we are interested in changes t
ward or away from a state of equilibrium, we c
use the equilibrium temperature T
, to expr
the temperature in a non-dimensional form. Thu
T+=TTi
TTi
(A
where
TTi=
V1(T1Ti)+V2(T2Ti)
1+V1+V2(A
is obtained from the steady-state solution of E
(A5).
Expressing the temperature T in Eq. (A5) terms of the dimensionless temperature T+ resu
in
~sdT+
dt=(1+V1+V2)(1T
+) (A
By defining the dimensionless time as
t+=(1+V1+V2)t
~s(A
we can transform Eq. (A8) into
dT+
dt=1T+ (A
which, with the initial condition
T+=0 at t+=0 (A
integrates into:
T+=1et+
(A
and, in view of Eqs. (A6) and (A9), can
transformed into:
TTi
TTi
=1e(1+V1+V2)~s (A
Referring to these results, we observe t
following.
(1) For one source only (say V1), Eq. (A
reduces to the equation cited in Bird et
(1960) (p. 490).
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N. Zuber /Nuclear Engineering and Design 204 (2001) 1 2726
(2) The method can be extended to any number
of sources and/or sinks.
(3) Given that
V1+V2=(1+2)~s (A14)
we may define the FRC for the system by
s=%i
i (A15)
and the effect metric for the system by
Vs=%i
Vi (A16)
Thus:
1+V1+V2=1+s~s=1+Vs (A17)
indicating that the FRC and the effect met-
rics V are additive.
(4)Consider a system transient induced by a set
of transfer processes. Given that, to each pro-
cess, there corresponds a metric V (which
quantifies its effect), the attendant set of V
metrics can be used to establish a hierarchy that
ranks the processes according to their impacton the transient. (The larger the effect metric V,
the more important the process.)
Such a quantitative, hierarchical ranking can be
used for both experimental and computer-based
studies.
For experiments, the hierarchy provides a
quantitative basis for establishing the design
and operation of a test facility. Specifically, it
identifies which effect metric V must be pre-
served in order to assure that a transfer process
will have the same effect in the prototype asthat observed in the test facility.
For computer-based analyses, the hierarchy es-
tablishes and ranks the modeling capabilities
that a code must have in order to assure its
applicability to a specific transient in a NPP.
(5)Consider now Eq. (A1). It has three design
parameters (V, A1, A2), two property parame-
ters (z, cp) and six operational parameters (WTin, h1, T1, h2, and T2).In the context of this particular example, tinformation of interest to a designer or toplant engineer would be concerned with teffects these parameters (and their variatiocan have on the fluid temperature in the vess
This information is contained in and providby a single relation (Eq. (A12)), made possibby scaling and expressing Eq. (A1) in a dimesionless form (Eq. (A10)).This simple example demonstrates the tributes of scaling in processing informatithrough the four stages represented in Fig.Specifically: the acquisition of information was effectiv
in as much as it required a single integrati(i.e. of Eq. (A10)) (simplicity);
the storage of information was optimal, inmuch as it is represented by a single curvethe T+t+ plane (synthesis and parsimony
the retrieval of information is fast and eain as much as it can be obtained fromsingle curve (efficiency);
the transmission of information is intellible, in as much as Eq. (A13) shows teffects of the various parameters (clarity)
(6) Consider now solving Eq. (A1) by means a computer code. In as much as this equation
cast in a dimensional form, to evaluate teffects of any parameter requires a separintegration (one for each variation).Such an acquisition of information, one piecea time, is ineffective, time consuming and epensive. Furthermore, the storage and retrievof information is most inefficient, given theach integration is displayed by a separcurve. Multiple curves are thereby generatwhich vary from facility to facility and froone test condition to another. This not onprecludes parsimony and synthesis, but it reders the transmission of information mdifficult, if not unintelligible.I hope that this simple example has served t
1. demonstrate the efficiency can be gainthrough scaling; and
2. illustrate the inefficiency and wastefulness the repetitive approach taken by computbased safety studies as presently conducted
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N. Zuber /Nuclear Engineering and Design 204 (2001) 1 27
Unfortunately, this piecemeal approach to ex-
periments and to computer-based analyses in
which the effects of various parameters are tested
and evaluated separately, one at a time, reflects a
cultural attitude that seems to permeate the cur-
rent T-H technology. Although cultures are not
easily changed, I see no reason to promulgate the
inane wastefulness of the code jockey attitude.
With regard to the latter, however, I am quite
optimistic. I am convinced that only those organi-
zations that stress efficiency in processing infor-
mation through the four stages of Fig. 1 will be
successful and will survive in a competitive envi-
ronment. Conversely, maladaptive organizations
that allow the code jockey culture to prevail, will
be inevitably relegated to the evolutionary junk
heap.
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