The Effects of Complexity

7/28/2019 The Effects of Complexity

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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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N. Zuber /Nuclear Engineering and Design 204 (2001) 1 272

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:


14/27


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).


15/27


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


16/27


17/27


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,


18/27


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


19/27


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


20/27


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.


21/27


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


22/27


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.


23/27


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.


24/27


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


25/27


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).


26/27


(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


27/27


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.

References

Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport

Phenomena. Wiley, New York, p. 216.

Blackmore, S., 1999. The Meme Machine. Oxford University

Press, Oxford.

Boyack, B.E., Catton, I., Duffey, R.B., Griffith, P., Katsina,

K.R., Lellouche, G.B., Levy, S., Rohatgi, U.S., Wilson,

G.E., Wulff, W., Zuber, N., 1990. An overview of the code

scaling, applicability and uncertainty evaluation methodol-

ogy. Nucl. Eng. Des. 119, 117.

Brodie, R., 1996. Virus of the Mind. Integral Press, Seattle,WA.

Bronowski, J., 1973. The Ascent of Man. Little, Brown and

Co., Boston, MA, pp. 155211.

Dawkins, R.D., 1976. The Selfish Gene. Oxford University

Press, Oxford, pp. 203215.

Ishii, M., Revankar, T., Leonardi, R., Dowlati, R., Bertodano,

M.S., Babelli, I.I., Wang, W., Pokharna, H., Ransom,

V.H., Viskanta, R., Han, J.T., 1998. The three-level scaling

approach with application to the Purdue University Multi-

Dimensional Integral Test Assembly (PUMA). Nucl. Eng.

Des. 186, 177211.

Jeffrey, W., Berger, J., 1992. Ockhams razor and Bayesian

analysis. Am. Sci. 80, 6472.Johnson, R.B., 1997. A Modern Introduction to the Mathe-

matical Theory of Water Waves. Cambridge University

Press, Cambridge, p. 98.

Kleiber, M., 1932. Body size and metabolism. Hilgardia 6,

315353.

Kolmogorov, A.X., 1941a. Dissipation of energy in locally

isotropic turbulence. Dokl. Akad. Nauk. SSSR 32, 1618

(in Russian)

Kolmogorov, A.X., 1941b. The local structure of turbulenc

incompressible viscous fluids of very large Reynolds nu

ber. Dokl. Akad. Nauk. SSSR 30, 913 (in Russian).

McMahon, Th. A., 1984. Muscles, Reflexes and Locomoti

Princeton University Press, Princeton, NJ, pp. 28329

Mills, R.I., 1994. Space, Time and Quanta. W.H. Freem

New York, pp. 89, 141.

Noether, E., 1918. Invariante variations probleme. Kgl. G

Wiss. Nachr. Gottingen Math. Phys. 2, 235 257.Oliver, O., 1994. The Shaggy Steed of Physics. Springer, N

York, pp. 3670.

Peterson, P.F., Schrock, V.E., Greif, R., 1998. Scaling

integral simulation of mixing in large, stratified volum

Nucl. Eng. Des. 186, 213224.

Stahl, W.R., 1967. Scaling of respiratory variables in ma

mals. J Appl. Physiol. 22, 453460.

Travkin, V.S., Catton, I., 2000. Transport phenomena

heterogeneous media based on volume averaging theo

In: Green, G. (Ed.), Advances in Heat Transfer, Vol.

Academic Press, New York, (in press).

Wilcox, D.C., 1998. Turbulence Modeling for CFD,

Canada, California, DCW Industries Inc., La CanaCalifornia, pp. 3639.

Witham, G.B., 1974. Linear and Nonlinear Waves. Wi

New York, p. 245.

Zuber, N., 1991. A hierarchical, two-tiered scaling analysis.

Technical Progress Group: An Integrated Structure a

Scaling Methodology for Severe Accident Technical Is

Resolution, Appendix D. NUREGICR 5809. US Nucl

Regulatory Commission.

Zuber, N., 1993. Two-phase systems in the context of co

plexity: a hierarchical approach for analysis and exp

ments. Invited Lecture at the ASME Fluids Engineer

Conference, Washington, DC, 2024 June 1993.

Zuber, N., 1994. Scaling and analysis of complex systeInvited Lecture at PSAIPAX Severe Accidents94, ES

and EXS, Ljubljana, Slovenia, 1720 April 1994.

Zuber, N., 1995. A general method for scaling and analyz

transport processes. Invited Lecture at the ASME-JSM

Annual Conference and Exhibit, Hilton Head, SC, 13

August 1995.

Zuber, N., 1999. A general method for scaling and analyz

transport processes. In: Lehner, M., Mewes, D., Tausch

R., Dinglreiter, U. (Eds.), Applied Optical Measuremen

Springer Verlag, Berlin.

Zuber, N., Wilson, G.E., Boyack, B.D., Catton, I., Duf

R.B., Griffith, P., Katsina, D.R., Lellouche, G.E., Lvey,

Rohatgl, U.S., Wulff, W., 1990. Evaluation of scalecapabilities of best estimate codes. Nucl. Eng. Des. 1

97109.

Zuber, N., Wilson, G., Ishii, M., Wulff, W., Boyack,

Dukler, A., Griffith, P., Healzer, J., Henry, R., Lehner

Levy, S., Moody, F., Pilch, M., Sehgal, B., Spencer,

Theofanous, T., Valente, J., 1998. An integrated struct

and scaling methodology for severe accident technical is

resolution Nucl Eng Des 186 122

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