A CENTURY OF TRANSPORT

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Feb 22, 2014 (3 years and 5 months ago)

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A CENTURY OF TRANSPORT


A Personal Tour


by


Stuart W. Churchill


DEPARTMENT OF CHEMICAL AND BIOMOLECULAR ENGINEERING


THE UNIVERSITY OF PENNSYLVANIA




TO REVIEW EVOLUTION OF THE SKILLS AND RESOURCES OF CHEMICAL
ENGINEERS IN DEALING WITH TRANSPORT




TO DESCRIBE NOT ONLY THE “STATE OF THE ART”, BUT ALSO TO TELL THE
STORY OF HOW WE GOT THERE




PREFERENCE IS BEING GIVEN TO THOSE PARTICULAR ASPECTS OF TRANSPORT
IN WHICH I HAVE BEEN INVOLVED


OBJECTIVES

WHAT IS TRANSPORT?


THE COMBINED TREATMENT OF FLUID MECHANICS, HEAT TRANSFER, AND MASS TRANSFER
AS
TRANSPORT

RATHER THAN AS SEPARATE TOPICS BECAME NOT ONLY FASHIONABLE BUT
ALSO THE GENERAL PRACTICE IN EDUCATION WITH THE PUBLICATION IN 1960 OF THE MOST
INFLUENTIAL BOOK IN THE HISTORY OF CHEMICAL ENGINEERING, NAMELY
TRANSPORT
PHENOMENA
BY BOB BIRD, WARREN STEWART, AND ED LIGHTFOOT.



ALTHOUGH OUR UNDERSTANDING OF TRANSPORT HAS EVOLVED OVER THE CENTURY AND
THE APPLICATIONS HAVE EXPANDED, THIS SUBJECT NOW HAS A DECREASED ROLE IN
EDUCATION AND PRACTICE BECAUSE OF COMPETITION FROM NEW TOPICS SUCH AS
BIOTECHNOLOGY AND NANOTECHNOLOGY. THESE LATTER TOPICS INVOLVE TRANSPORT BUT
MOSTLY AT SUCH A SMALLER SCALE THAT WHAT I WILL BE DESCRIBING IS APPLICABLE, IF AT
ALL, ONLY IN A QUALITATIVE SENSE OR AS A GUIDE TO THE DEVELOPMENT OF EQUIVALENT
RELATIONSHIPS
.

CONTINUITY AND CONSERVATION


THE EQUATIONS OF CONSERVATION
-

THE NAVIER STOKES EQUATIONS AND THEIR
COUNTERPARTS FOR ENERGY AND SPECIES
-

ARE THE STARTING POINT OF MOST
THEORETICAL WORK ON TRANSPORT. I WILL NOT TRACE THE DEVELOPMENT OF THESE
EQUATIONS NOR EXAMINE THEIR VALIDITY EXCEPT TO CITE ONE CONTRARY OPINION FROM
A RENOWNED PHYSICIST.



GEORGE E. UHLENBECK, ONE OF MY TEACHERS AND MENTORS, FRUSTRATED BY HIS FAILURE
TO CONFIRM OR DISPROVE THE NAVIER
-
STOKES EQUATIONS BY REFERENCE TO STATISTICAL
MECHANICS, WHICH HE CONSIDERED TO BE A BETTER STARTING POINT, ONCE WROTE THE
FOLLOWING:


“QUANTITATIVELY, SOME OF THE PREDICTIONS FROM

THESE EQUATIONS SURELY DEVIATE FROM EXPERIMENT,

BUT THE
VERY
REMARKABLE FACT REMAINS
THAT

QUALITATIVELY
THE NAVIER
-
STOKES EQUATIONS

ALWAYS


DESCRIBE PHYSICAL PHENOMENA SENSIBLY.

THE MATHEMATICAL REASON FOR THIS VIRTUE OF THE

NAVIER
-
STOKES EQUATIONS IS COMPLETELY MYSTERIOUS

TO ME.”


CONCEPTUAL AND COMPOUND VARIABLES



SOME OF UNIQUE CONCEPTS AND COMPOUND VARIABLES OF TRANSPORT HAVE BECOME
SO COMMONPLACE THAT WE MAY NO LONGER APPRECIATE HOW INVALUABLE THEY ARE,
OR REMEMBER WHERE THEY CAME FROM AND THEIR LIMITS OF VALIDITY.



I WILL CALL TO YOUR ATTENTION A FEW OF THEM :

1) THE HEAT TRANSFER COEFFICIENT AND ITS ANALOGUES

2) THE EQUIVALENT THICKNESS FOR PURE CONDUCTION






5) MIXED
-
MEANS IN GENERAL


6) FULLY DEVELOPED FLOW

7) THE FRICTION FACTOR FOR ARTIFICIALLY ROUGHENED TUBES

8)

THE FRICTION FACTOR FOR COMMERCIAL (NATURAL) ROUGHNESS

9) THE EQUIVALENT LENGTH

10) “PLUG FLOW”

11) INTEGRAL BOUNDARY
-
LAYER THEORY


)
/
2
(
1
ln{
/
2



1912
in
Langmuir
f
Nu
Nu
















1
0
2

ity
mean veloc
-
mixed

The

3)
a
r
d
a
r
u
u
r
m






















1
0
2

rature
mean tempe
-
mixed

The

4)
a
r
d
a
r
u
u
T
T
m
r
r
m
12) POTENTIAL FLOW AND THE THIN
-
BOUNDARY
-
LAYER CONCEPT

13) FREE STREAMLINES PREDICT = 0.611 FOR ORIFICE. THE COEFFICIENTREAL VALUE IS

0.5793.

14) CRITERIA FOR TURBULENT FLOW IN PIPES: OSBORNE REYNOLDS IN 1883












REYNOLDS:

Re

= 2100

OR
a
+

=

a
(
τ
w

ρ
)
½

/
μ

=
Re(f/
8)
½
56



MODERN: LAMINAR
: a
+




45 [
Re

≤ 1600]; TURBULENT: a
+


≥ 150 [
Re

≥ 4020]





2
15) FULLY
-
DEVELOPED CONVECTION



UNIFORM HEATING




THE NEAR
-
ATTAINMENT OF ASYMPTOTIC VALUES OF (
T−T
0
)/(
T
m
−T
0
) AS A

FUNCTION
OF
r/a

AND OF THE LOCAL HEAT TRANSFER COEFFICIENT




UNIFORM WALL
-
TEMPERATURE




THE NEAR
-
ATTAINMENT OF ASYMPTOTIC VALUES OF (
T
w
−T
)/(
T
w
−T
m
)

AS A
FUNCTION OF
r/a

AND OF THE LOCAL HEAT TRANSFER COEFFICIENT

16) THE BOUSSINESQ TRANSFORMATION



MOST NOTABLY THE REPLACEMENT OF


g



(∂
p/

x
)/
ρ

BY


(
T


T

)

17) THE RADIATIVE HEAT TRANSFER COEFFICIENT





LINEARIZATION ALLOWS USE WITH OHM’S LAWS

18) BLACK
-
BODY AND GRAY
-
BODY RADIATION


3
2
4
4
4
4
)
)(
(
)
(
m
s
s
s
s
s
R
T
T
T
T
T
T
T
T
T
h













19) ASYMPTOTIC SOLUTIONS FOR TURBULENT FREE CONVECTION



NUSSELT IN 1915
:
h
APPROACHES INDEPENDENT FROM

x

AS

x






REQUIRES
Nu
x


Gr
x
1/3




FRANK
-
KAMENETSKII IN 1937:

h

INDEPENDENT OF

k
AND
μ




REQUIRES

Nu
x

Gr
x
1/2

Pr


ECKERT AND JACKSON

IN 1951: INTEGRAL BOUNDARY LAYER THEORY


Nu
x

Gr
x
0.4





CHURCHILL IN 1970
:
Nu
x


A

Ra
x
1/3

AS

Pr

→ ∞

AND

x





Nu
x



B
(
Ra
x
Pr
)
1/3

AS
Pr


0

AND

x






SEEMINGLY

VALIDATED BY LIMITED EXPERIMENTAL DATA


20) OHM’S DERIVED IN 1827 EXPRESSIONS FOR STEADY
-
STATE ELECTRICAL CONDUCTION
REGULARLY APPLIED IN CHEMICAL ENGINEERING FOR OTHER LINEAR BEHAVIOR






SPECIAL FORMS OF TRANSPORT

1) FLUIDIZED BEDS: THE ALMOST EXCLUSIVE DOMAIN OF CHEMICAL ENGINEERS


DICK WILHELM AND MOOSUN KWAUK IN 1948




1) INCIPIENT FLUIDIZATION: −∆
P = L
(1−

ε
)
g
(
ρ
s


ρ
)


2) HEIGHT OF EXPANDED BED:

L
(1−

ε
)

=
L

(1−

ε
)


3) MEAN INTERSTITIAL VELOCITY:


u
m0

= u
T

ε
n



AFTER MORE THAN 60 YEARS, FLUIDIZATION IS STILL A LIVELY SUBJECT OF




RESEARCH

2) PACKED BEDS



MAJORITY OF CONTRIBUTIONS HAVE BEEN BY CHEMICAL ENGINEERS, AGAIN




BECAUSE OF THE APPLICABILITY TO CATALYSIS


EARLY EXAMPLE


SABRI ERGUN IN 1952:



75
.
1
)
1
(
150
1
9
2
0
3




















u
D
u
D
L
P
p
p
3) LAMINAR CONDENSATION


NUSSELT IN 1916 FOR A FILM FALLING DOWN A VERTICAL PLATE:








HERE,
Г
IS THE MASS RATE OF CONDENSATION PER UNIT BREADTH

SEVERAL YOU MAY NOT KNOW ABOUT

4) MIGRATION OF WATER IN POROUS MEDIA


MEASUREMENTS BY JAI P. GUPTA OF THE WATER CONCENTRATION IN SAND DURING


FREEZING AT A SUBCOOLED SURFACE REVEALED THAT WATER MIGRATES TO THE


FREEZING FRONT FASTER THAN CAN BE EXPLAINED BY DIFFUSION. THE VARIATION



OF SURFACE TENSION WITH TEMPERATURE WAS FOUND TO BE THE CAUSE.

5) CONVECTION DRIVEN BY A MAGNETIC FIELD


STUDIED IN DEPTH AND ALMOST EXCLUSIVELY BY HIROYUKI OZOE. APPLICATIONS:


ZOCHRALSKI CRYSTALIZATION AND SEPARATION OF GASES IN SPACE VEHICLES



AND STATIONS.




9428
.
0
3
4
4
/
3
4
/
1
3
3
3
2
3












L
T
T
k
g
p
g






6) THERMOACOUSTIC CONVECTION



INCORPORATION OF FOURIER’S EQUATION IN THE UNSTEADY
-
STATE, ONE
-
DIMENSIONAL
DIFFERENTIAL ENERGY BALANCE RESULTS IN:




MATHEMATICIANS HAVE LONG RECOGNIZED THAT THIS MODEL PREDICTS AN INFINITE RATE OF
PROPAGATION OF ENERGY.



CATTANEO IN 1948, MORSE AND FESHBACH IN 1953, AND VERNOTTE IN 1958
INDEPENDENTLY PROPOSED THE SO
-
CALLED
HYPERBOLIC EQUATION OF CONDUCTION

TO AVOID
THAT DEFECT
:






HERE,
u
T

IS THE VELOCITY OF A THERMAL WAVE. THIS CONCEPT IS PURE RUBBISH!




NUMERICAL SOLUTIONS OF THE EQUATIONS OF CONSERVATION AND EXPERIMENTAL
MEASUREMENTS BY MATTHEW BROWN CONFIRMED OUR CONJECTURE THAT THE WAVE IS
GENERATED BY COMPRESSIBILITY WITHOUT THE NEED FOR ANY SUCH A HEURISTIC.
















x
T
k
x
t
T
c

















x
T
k
x
t
T
u
k
t
T
c
T
2
2
2

7) THERMAL CONDUCTION THROUGH DISPERSIONS





MAXWELL IN 1873, USING THE PRINCIPLE OF INVARIANT IMBEDDING, DERIVED AN
APPROXIMATE SOLUTION FOR THE ELECTRICAL CONDUCTIVITY OF DISPERSIONS OF SPHERES.




IN 1986 I FOUND THAT, WHEN RE
-
EXPRESSED IN THERMAL TERMS AND RE
-
ARRANGED IN TERMS OF ONE DEPENDENT AND ONE AND INDEPENDENT VARIABLE, THIS
SOLUTION PROVIDED A

LOWER BOUND AND A FAIR REPRESENTATION EVEN FOR THE
EXTREME OF A PACKED BED AND EVEN FOR GRANULAR MATERIALS.

SIMILARITY TRANSFORMATIONS

A FEW FAMILIAR EXAMPLES

1) TRANSIENT THERMAL CONDUCTION

2) THE THIN BOUNDARY
-
LAYER TRANSFORMATION

OF PRANDTL IN 1904


3) THE POHLHAUSEN TRANSFORMATION OF 1921

FOR FREE CONVECTION

4) THE LÉVÊQUE TRANSFORMATION OF 1928



5) THE INTEGRAL TRANSFORMATION OF DUDLEY A. SAVILLE IN 1967 FOR FREE


CONVECTION



THE HELLUMS
-
CHURCHILL METHODOLOGY OF 1964



COMPUTERIZED IN 1981 BY CHARLES W. WHITE, III


CONVENTIONAL CORRELATING EQUATIONS


POWER
-
LAW RELATIONSHIPS BASED ON LOGARITHMIC PLOTS OF DIMENSIONLESS GROUPS




SCATTER IS USUALLY DUE TO:




1) UNRECOGNIZED PARAMETERS




2) WRONG CHOICE OF DIMENSIONLESS GROUPINGS




3) NON
-
LOGARITHMIC DEPENDENCE



A CLASSICAL EXAMPLE FOLLOWS:

DIMENSIONAL ANALYSIS OF A LIST OF VARIABLES


RAYLEIGH HAD “THE LAST WORD” WHEN IN 1915 HE DERIVED


Nu = A Re
n

Pr
m

+
B Re
2n

Pr
2m

+
Re
3n

Pr
4m

+.....


HE EMPHASIZED THAT THIS ONLY MEANT THAT


Nu = Φ
{
Re, Pr
}


SUBSEQUENT “CONTRIBUTIONS TO DIMENSIONAL ANALYSIS” ARE BEST IGNORED


INFERENCES



POWER
-
DEPENDENCES OCCUR ONLY FOR ASYMPTOTIC BEHAVIOR


WE SHOULD STOP DRAWING LINES THROUGH SCATTERED DATA ON LOG
-



LOG PLOTS

A CORRELATING EQUATION FOR ALMOST EVERYTHING



IN 1972 WE BEGAN TESTING AS A GENERAL EXPRESSION FOR CORRELATION:




WE CALLED THIS THE CHURCHILL

USAGI EQUATION OR CUE.



THE INCORPORATION OF ASYMPTOTES IMPROVED ACCURACY BOTH

NUMERICALLY AND FUNCTIONALLY BEYOND ALL EXPECTATIONS

.


WE WERE NOT THE FIRST TO UTILIZE THIS EXPRESSION:

EARLIER USERS

INCLUDE ANDY
ACRIVOS AND TOM HANRATTY.


OUR CONTRIBUTIONS WERE:




1) TO RECOGNIZE ITS FULL POTENTIAL




2) TO DEVISE AN OPTIMAL PROCEDURE FOR DETERMINATION OF THE ARBITRARY
EXPONENT
n

BASED ON THE

ALTERNATIVE FORMS:





p
p
p
x
y
x
y
x
y
/
1
0
0

}
{
}
{
}
{




}
{
}
{
1
}
{
}
{
/
1
0
0
n
n
x
y
x
y
x
y
x
y



















n
n
x
y
x
y
x
y
x
y
/
1
0
}
{
}
{
1
)
(
}
{





















AND





OUR FIRST APPLICATION
-

LAMINAR FREE CONVECTION

FROM AN ISOTHERMAL VERTICAL
PLATE IN THE THIN LAMINAR BOUNDARY LAYER REGIME
-

RESULTED IN:









GRAPHICAL EVALUATION OF
n

:





n
n
x
Gr
x
Gr
n
x
Nu














4
/
1
Pr
5027
.
0
2
/
1
Pr
6004
.
0


9
/
4
16
/
9
4
/
1
Pr
492
.
0
1
Pr)
5072
.
0
















x
x
Gr
Nu
FOR
n =

9/4, PER THE GRAPH:

A SUBSEQUENT EARLY APPLICATION

THE VELOCITY DISTRIBUTION IN TURBULENT FLOW IN A ROUND TUBE


ASYMPTOTES








COMBINATION











10
30
ln
5
.
2
5
.
5

and

0

as















a
y
for
y
u
y
y
u


n
n
y
y
u
y























ln
5
.
2
5
.
5
1
THE CANONICAL PLOT


THE CONVENTIONAL PLOT OF THE SAME VARIABLES

RESTRICTIONS ON THE CUE


ASYMPTOTES MUST BE KNOWN, DERIVED, OR FORMULATED



ASYMPTOTES MUST INTERSECT ONCE AND ONLY ONCE



ASYMPTOTES MUST BOTH BE UPPER BOUNDS OR LOWER BOUNDS



ASYMPTOTES MUST BOTH BE FREE OF SINGULARITIES



BEHAVIOR MUST BE REASONABLY SYMMETRICAL WITH RESPECT TO THE ASYMPTOTES
(CANNOT EXPECT TO BE FULFILLED EXACTLY)

GUIDELINES


DIFFERENTIATION AND INTEGRATION LEAD TO AWKWARD EXPRESSIONS.


DIFFERENTIATE OR INTEGRATE ASYMPTOTES AND DEVISE A SEPARATE


CORRELATING EQUATION WITH A DIFFERENT COMBINING EXPONENT.


STATISTICAL ANALYSIS IS UNNECESSARY:


THE EXPRESSION IS SO INSENSITIVE TO THE VALUE OF
n
THAT A RATIO


OF INTEGERS MAY BE CHOSEN.


ELI RUCKENSTEIN DERIVED A THEORETICAL VALUE OF 3 FOR
n
FOR FREE AND



FORCED CONVECTION. THIS VALUE HOLDS FOR MOST OTHER


COMBINATIONS OF ASSISTING OR OPPOSING MECHANISMS.


IN SOME INSTANCES, A THEORETICAL RATIONLIZATION EXISTS FOR
n =
1
OR n =

1.

MULTIPLE VARIABLES



MAY BE INCORPORATED IN ASYMPTOTES AS IS
Gr
X

IN THE PRIOR EXAMPLE, NAMELY:








MAY BE INTRODUCED

SERIALLY, AS IN:




9
/
4
16
/
9
4
/
1
Pr
492
.
0
1
Pr
5027
.
0

















X
X
Gr
Nu


m
n
m
n
n
m
y
y
y
y




/
1
0
TRANSITIONAL BEHAVIOR


REQUIRES SPECIAL MEASURES


THE INTERMEDIATE (TRANSITIONAL) ASYMPTOTE IS SELDOM KNOWN BUT CAN ALMOST
ALWAYS BE REPRESENTED BY AN ARBITRARY POWER LAW.


DIRECT SERIAL APPLICATION FAILS IF
y
0

IS A

LOWER AND

y


AN UPPER BOUND, AND VICE
VERSA.

THIS ANOMALY CAN BE AVOIDED BY USING
“STAGGERED” VARIABLES SUCH AS:







WHICH FOLLOWS FROM

APPLICATION OF THE
CUE TO
y
0

AND
y
1

, NAMELY:






AND THEN IN TURN TO
y

,

NAMELY:


n
n
n
y
y
z
0


n
n
n
y
y
y
1
0






m
n
n
nm
m
n
n
y
y
y
y
y
0
1
0






A SPECIFIC APPLICATION OF “STAGGERING” IS PROVIDED BY THE INDICATED EXPRESSION
FOR THE EFFECTIVE VISCOSITY OF A PSEUDOPLASTIC












THIS PROCESS AND RESULT SUGGESTS THE POWER
-
LAW MAY BE A MATHEMATICAL
ARTIFACT


A GENERALIZED REPRESENTATION FOR TRANSITION





HICKMAN IN 1974 CARRIED OUT NUMERICAL CALCULATIONS FOR A SERIES OF BIOT
NUMBERS.



HIS RESULTS AND CORRELATION CAN BE RE
-
EXPRESSED IN TERMS OF THE CUE AS:





HERE, THE SUBSCRIPTS J AND T DESIGNATE UNIFORM AND ISOTHERMAL HEATING OR
COOLING, BUT

THIS EXPRESSION CAN BE ADAPTED AS A GENERALIZED ONE FOR ALL
TRANSITIONAL PROCESSES.

T
T
T
J
Nu
Bi
Nu
Nu
Nu
Nu




1
THE STATUS AND FUTURE OF THE CUE

ANALOGIES


HAVE A PERVASIVE ROLE IN CHEMICAL ENGINEERING



EXAMPLES:



THE EQUIVALENT DIAMETER

(THE CHOICE IS NOT UNIQUE)


THE ANALOGY OF MACLEOD


THE ANALOGY BETWEEN HEAT AND MASS TRANSFER (TO BE EXAMINED

IN
DETAIL SUBSEQUENTLY)


THE ANALOGY BETWEEN ELECTRICAL AND THERMAL CONDUCTION


THE ANALOGY OF EMMONS FOR ALL BUOYANT PROCESSES (FREE

CONVECTION, FILM
CONDENSATION, FILM BOILING, AND FILM

MELTING)


4
/
1
3

LAMINAR










T
k
K
FL
k
L
h
m



3
/
1
3

TURBULENT










T
k
K
FL
A
k
L
h
m



A NEW ANALOGY BETWEEN CHEMICAL REACTION AND CONVECTION


THE RADICAL ENHANCEMENT AND ATTENUATION OF CONVECTION BY ENERGETIC
CHEMICAL REACTIONS HAVE BEEN KNOWN FOR OVER 40 YEARS BUT IS NOT EVEN
MENTIONED IN TEXTBOOKS.



EARLIEST INVESTIGATORS INCLUDE THIBAULT BRIAN, BOB REID, AND SAMUEL BODMAN IN
THE PERIOD 1961
-
1965, JOE SMITH IN 1966, AND LOUIS EDWARDS AND ROBERT
FURGASON IN 1968.



WHILE MODELING COMBUSTION IN 1972 I BECAME AWARE OF THIS EFFECT, AND MANY
YEARS LATER DERIVED THE FOLLOWING:





















1
0
0
/
1
/
exp
1
1






x
mx
mx
ox
x
K
Z
RT
E
Z
Nu
Nu


THIS EQUATION MAY BE INTERPRETED AS AN ANALOGY RELATING


THE LOCAL RATE OF HEAT TRANSFER, AS REPRESENTED BY

Nu
x

,


TO THE LOCAL MIXED
-
MEAN RATE OF REACTION AS REPRESENTED


BY .



mx
Z

1

ILLUSTRATIVE REPRESENTATIONS



LAMINAR FLOW


















HERE
K
0x

= k
0x
/u
m

IS THE DIMENSIONLESS DISTANCE THROUGH THE REACTOR



TURBULENT FLOW

TURBULENT FLOW



FOR OVER HALF OF OUR CENTURY, PRANDTL AND HIS STUDENTS, COLLEAGUES, AND
CONTEMPORARIES UTILIZED DIMENSIONAL AND SPECULATIVE ANALYSIS TO DEVISE AN INGENIOUS
STRUCTURE FOR THE THEN
-
INTRACTABLE PROCESS OF TURBULENT FLOW.



ONE OF THEIR IMPRESSIVE CHARACTERISTICS WAS RESILIANCE; IF ONE APPROACH WAS FOUND
TO BE FLAWED, THEY TRIED ANOTHER AND ANOTHER.

TIME
-
AVERAGING OF THE EQUATIONS OF CONSERVATION



OSBORNE REYNOLDS IN 1895

SPACE
-
AVERAGED THESE EQUATIONS FOR A ROUND TUBE



THIS WAS THE GREATEST SINGLE ADVANCE OF ALL TIME IN TURBULENT FLOW.

THE EDDY DIFFUSIVITY

CONCEIVED OF BY BOUSSINESQ IN 1877

THE POWER LAW FOR THE FRICTION FACTOR



BLASIUS IN 1913 INFERRED FROM EXPERIMENTAL DATA

THAT
f
WAS INVERSELY PROPORTIONAL
TO

Re
1/4.




UNFORTUNATELY, THIS IS A CRUDE APPROXIMATION THAT DOES NOT APPLY TO ANY FINITE
RANGE OF
Re.

THE POWER LAW FOR THE VELOCITY DISTRIBUTION


PRANDTL IN 1921 RECOGNIZED THAT THE POWER
-
LAW OF BLASIUS FOR THE FRICTION
FACTOR REQUIRED:



HE ALSO RECOGNIZED ITS FAILURE IN BOTH LIMITS FOR ANY EXPONENT.

WALL
-
BASED VARIABLES



PRANDTL IN 1926 USED DIMENSIONAL ANALYSIS TO DERIVE:







THESE DIMENSIONLESS VARIABLES AND SYMBOLS HAVE REMAINED IN ACTIVE AND
PRODUCTIVE USE FOR OVER 80 YEARS.

THE UNIVERSAL LAW OF THE WALL


PRANDTL NEXT CONJECTURED THAT NEAR THE WALL THE DEPENDENCE ON
a+
SHOULD
PHASE OUT LEADING TO: .


7
/
1







a
r
u
u
c
r

















a
y
u
a
y
u
w
w
w
,

or

,













y
u




THE UNIVERSAL LAW OF THE CENTER


PRANDTL SIMILARLY CONJECTURED THAT THE VELOCITY FIELD NEAR THE CENTERLINE
MIGHT BE INDEPENDENT OF THE VISCOSITY LEADING TO:


THE MIXING LENGTH

CONCEIVED BY PRANDTL IN 1925


THE SEMI
-
LOGARITHMIC VELOCITY DISTRIBUTION


THE CONJECTURE OF PRANDTL THAT NEAR THE WALL THE MIXING LENGTH WOULD DEPEND
LINEARLY ON THE DISTANCE FROM THE WALL (NAMELY THAT
l = ky
) LEAD HIM TO:
.


THE 3/2
-
POWER EXPRESSION FOR THE VELOCITY DEFECT



PRANDTL IN 1925 FURTHER CONJECTURED THAT THE MIXING LENGTH MIGHT APPROACH A
CONSTANT VALUE AT THE CENTERLINE LEADING TO THE FOLLOWING ERRONEOUS
EXPRESSION: .




y
a
u
u
c
/





}
ln{
1




y
k
A
u
2
/
3
Br
u
u
c


AN OVERALL EXPRESSION FOR THE MIXING
-
LENGTH



IN 1930, IN ORDER TO ENCOMPASS A WIDER RANGE OF BEHAVIOR, VON KÁRMÁN
PROPOSED:


A SEMI
-
LOGARITHMIC EXPRESSION FOR THE MIXED
-
MEAN VELOCITY AND THE FRICTION
FACTOR


VON KÁRMÁN AND PRANDTL INDEPENDENTLY CONJECTURED THAT, IN SPITE OF ITS
FAILURES NEAR THE WALL AND NEAR THE CENTERLINE, THE INTEGRATION OF THE SEMI
-
LOGARITHMIC EXPRESSION FOR THE VELOCITY OVER THE CROSS
-
SECTION MIGHT YIELD A
GOOD APPROXIMATION FOR THE MIXED
-
MEAN VELOCITY AND THEREBY THE FRICTION
FACTOR, NAMELY

:










2
2
/
/
dy
u
d
dy
du
k

}
ln{
1
}
ln{
1
2
3

2










a
k
B
a
k
k
A
f
u
u
w
m
m

AN IMPROVED DERIVATION OF THE SEMI
-
LOGARITHMIC VELOCITY DISTRIBUTION



MILLIKAN IN 1938 RECOGNIZED THAT THE ONLY EXPRESSION CONFORMING TO BOTH “THE
LAW OF THE WALL” AND “THE LAW OF THE CENTER” WAS:



THIS ALTERNATIVE DERIVATION OF “THE LAW OF THE TURBULENT CORE NEAR THE WALL,”
WHICH IS FREE OF ANY HEURISTICS, REVEALS THAT TWO ERRONEOUS CONCEPTS (THE
MIXING LENGTH AND ITS LINEAR VARIATION NEAR THE WALL) FORTUITOUSLY LED TO A
VALID RESULT.

THE LINEAR VELOCITY DISTRIBUTION VERY NEAR THE WALL


PRANDTL POSTULATED THAT VERY, VERY NEAR THE WALL THE SHEAR STRESS DUE TO THE
TURBULENT FLUCTUATIONS AND THE EFFECT OF CURVATURE WOULD BE EXPECTED TO BE
NEGLIGIBLE, LEADING TO:






THIS EXPRESSION CAN BE NOTED TO CONFORM TO “THE LAW OF THE WALL.


}
ln{
1




y
k
B
u












y
a
y
y
u
2
1
THE TURBULENT SHEAR STRESS VERY NEAR THE WALL


IN 1932, EGER MURPHREE, A CHEMIST, AND SOMEWHAT LATER, CHARLIE WILKIE, A
CHEMICAL ENGINEER, AND HIS ASSOCIATES PROPOSED THAT:






THE EXISTENCE OR NON
-
EXISTENCE OF

THE TERM IN (
y
+
)
3
WAS DISPUTED FOR OVER 50
YEARS.



THIS ISSUE WAS FINALLY SETTLED DEFINITIVELY BY THE RESULTS OF

DNS
, INCLUDING THOSE
OF RUTLEDGE AND SLEICHER, AND OF LYONS, HANRATTY, AND MCLAUGHLIN, WHICH ALSO
DETERMINED
α ≈

0.00700.








4
3
)
(
)
(
'
'
y
y
v
u



POST
-
PRANDTL MODELING

THE
k
-
ε

MODEL



FOLLOWS FROM THE CONJECTURES OF KOLMOGOROV, PRANDTL, AND BATCHELOR


EMPIRICAL EQUATIONS FOR
k
AND
ε

WERE DEVISED BY LAUNDER AND SPALDING IN 1972.


THE PREDICTIONS OF FLOW NEAR THE WALL REMAIN POOR.


IT IS NEVERTHELESS OUR BEST RESOURCE FOR MODELING DEVELOPING FLOW.

DIRECT NUMERICAL SIMULATION (
DNS
)



CHARLES SLEICHER AND TOM HANRATTY AND THEIR DOCTORAL STUDENTS FOLLOWED THE LEAD
OF KIM, MOIN AND MOSER IN 1987 AND USED
DNS
TO PREDICT TURBULENT FLOW IN PARALLEL
-
PLATE CHANNELS.


NUMERICAL SOLUTIONS ARE STILL LIMITED TO RATES OF FLOW JUST ABOVE THE MINIMUM FOR
FULLY DEVELOPED TURBULENCE, NAMELY,
Re =

4000.


DNS

REQUIRES EXCESSIVE COMPUTATION FOR ROUND TUBES OR ANNULI.


LARGE
-
EDDY SIMULATION (LES)



THIS MODEL, AS DEVISED BY SCHUMANN IN 1975, RELAXES THE RESTRICTION ON THE RATE
OF FLOW BY UTILIZING
DNS

ONLY FOR THE FULLY TURBULENT CORE, BUT IS INACCURATE
NEAR THE WALL BECAUSE OF THE USE OF THE
k
-
ε

MODEL WITH ARBITRARY WALL
-
FUNCTIONS
.


THE FUTURE OF NUMERICAL SIMULATION


WE SORELY NEED A NEW ALGORITHM OR CONCEPT THAT WILL EXTEND THE PREDICTIONS
OF TURBULENT FLOW TO ROUND TUBES AND LARGE REYNOLDS NUMBERS, AS PROMISED
BUT NOT DELIVERED BY
DNS

AND
LES.

THE LOCAL FRACTION OF THE SHEAR STRESS DUE TO TURBULENCE



IN 1995, CHRISTINA CHAN AND I PROPOSED THE DIRECT CORRELATION OF EXPERIMENTAL
AND COMPUTED VALUES FOR THE TURBULENT SHEAR STRESS, THEREBY AVOIDING THE
HEURISTICS SUCH AS THE EDDY VISCOSITY AND THE MIXING LENGTH.


OUR FIRST CHOICE OF A DIMENSIONLESS VARIABLE WAS:



WE SUBSEQUENTLY PROPOSED THE FOLLOWING IMPROVED ONE, WHICH IS FINITE AT THE
CENTERLINE:




IS SEEN TO BE THE LOCAL FRACTION OF THE SHEAR STRESS DUE TO THE TURBULENT
FLUCTUATIONS.


IT IS WELL
-
BEHAVED FOR ALL CONDITIONS AND, IN CONTRAST TO , IS FINITE AT THE
CENTERLINE.

w
v
u
v
u


/
)
'
'
(
)
'
'
(





/
)
'
'
(
)
'
'
(
v
u
v
u






)
'
'
(
v
u

)
'
'
(
v
u


IT IS EASY TO SHOW THAT:




THIS RESULT CONFIRMS THAT, DESPITE ITS HEURISTIC ORIGIN AND THE CONTEMPT OF MANY “PURISTS,” THE
EDDY VISCOSITY REALLY HAS SOME PHYSICAL SIGNIFICANCE.



AT THE SAME TIME, THE EDDY VISCOSITY IS INFERIOR TO IN TERMS OF SIMPLICITY AND
SINGULARITIES, AND IS THEREFORE NOW OF HISTORICAL INTEREST ONLY.


THE EXPRESSION FOR THE MIXING LENGTH REVEALS THAT IT IS INDEPENDENT OF ITS MECHANISTIC AND
HEURISTIC ORIGIN. HOWEVER, IT IS ALSO REVEALED TO BE UNBOUNDED AT THE CENTERLINE OR THE
CENTRAL PLANE OF A PARALLEL PLATE CHANNEL.


HOW DID SUCH AN ANOMALY ESCAPE ATTENTION FOR MORE THAN 70 YEARS?

ONE EXPLANATION IS THE
UNCRITICAL ACCEPTANCE BY PRANDTL OF THE PLOT OF VALUES OF THE MIXING LENGTH OBTAINED FROM
THE “ADJUSTED” EXPERIMENTAL VALUES OF NIKURADSE, FOLLOWED BY THE UNCRITICAL EXTENSION OF
RESPECT FOR PRANDTL AND VON KÁRMÁN TO ALL OF THEIR DERIVATIONS.









2
2
'
'
1
1
'
'

and

)
'
'
(
1
)
'
'
(




















v
u
a
y
v
u
v
u
v
u
t





)
'
'
(
v
u
AN ALGEBRAIC CORRELATING EQUATION FOR THE TURBULENT SHEAR STRESS


IN 2000 WE DEVISED, USING THE
CUE
,
THE FOLLOWING THEORETICALLY
-
BASED EXPRESSION FOR THE LOCAL
FRACTION OF THE TOTAL SHEAR STRESS DUE TO TURBULENCE:



THIS EXPRESSION COMBINES ASYMPTOTES FOR THREE REGIONS AND THE LATEST EXPERIMENTAL DATA FOR
u
+

AS WELL AS FOR .


ACCORDING TO THE ANALOGY OF MCLEOD, THIS EXPRESSION IS APPLICABLE FOR PARALLEL

PLATE CHANNELS
IF
b
+

IS SUBSTITUTED FOR
a
+
. WE HAVE ALSO ADAPTED IT FOR CIRCULAR CONCENTRIC ANNULI.


THE ULTIMATE PREDICTIVE EQUATION FOR THE FRICTION FACTOR IN A ROUND TUBE IS:




AN ITERATIVE SOLUTION IS REQUIRED TO DETERMINE THE FRICTION FACTOR FOR A SPECIFIED VALUES OF
Re =

2
a
+
u
m
+

AND
e/a,
BUT CONVERGENCE IS VERY RAPID.

8
/
7
7
/
8
7
/
8
3
95
.
6
1
436
.
0
1
436
.
0
1
exp
10
7
.
0
)
'
'
(























































a
y
a
y
y
v
u


)
'
'
(
v
u























a
a
e
a
a
a
f
u
m
)
/
(
301
.
0
1
ln
436
.
0
1
50
227
3
.
3
2
2

THE CORRESPONDING EXPRESSION FOR THE FRICTION FACTOR OF ALL REGIMES OF FLOW
(LAMINAR, TRANSITIONAL, AND TURBULENT) AND ALL EFFECTIVE ROUGHNESS RATIOS IS:






HERE,
f
l

= 16/
Re
(POISEUILLE’S LAW),
f
t

=
(
Re/
37530)
2
,
AND

f
T

IS THE ABOVE EXPRESSION FOR
FULLY TURBULENT FLOW
.

THIS EXPRESSION IS A COMPLETE REPLACEMENT FOR AND
IMPROVEMENT ON ALL EXPRESSIONS AND PLOTS FOR THE FRICTION FACTOR.


ALTHOUGH IT OBVIATES THE NEED FOR ONE, IT IS CAN READILY BE PROGRAMMED TO
PRODUCE SUCH A PLOT IN EVERY DETA.


EXPERIMENTAL DATA FOR TURBULENT FLOW OF GREATEST HISTORICAL SIGNIFICANCE

BLASIUS IN 1913

NIKURADSE IN 1930, 1932, and 1933

COLEBROOK IN 1938
-
1939

ZAGAROLA IN 1996

12
/
1
2
/
3
16
16
12
)
]
[
(






T
t
l
f
f
f
f
TURBULENT CONVECTION



UNFOLDS PRIMARILY THROUGH ANALOGIES BETWEEN MOMENTUM AND ENERGY
TRANSFER.


THE SOLUTION OF SLEICHER IN 1956, USING AN ANALOG COMPUTER, IS A PARTIAL
EXCEPTION; IT WAS UPGRADED IN 1969 BY NOTTER AND SLEICHER USING A DIGITAL
COMPUTER.


A GENERALIZED CORRELATING EQUATION FOR FORCED CONVECTION


IN 1977, I DEVISED, USING THE

CUE
WITH 5 ASYMPTOTES AND FOUR COMBINING
EXPONENTS, A CORRELATING EQUATION FOR
Nu

FOR ALL
Pr

AND ALL
Re
(INCLUDING THE
LAMINAR, TRANSITIONAL, AND TURBULENT REGIMES). THE SAME STRUCTURE BUT
DIFFERENT ASYMPTOTES WERE PROPOSED FOR UNIFORM HEATING AND UNIFORM WALL
TEMPERATURE. THESE EXPRESSIONS ARE HERE COMPARED GRAPHICALLY WITH
EXPERIMENTAL DATA AND A FEW NUMERICALLY COMPUTED VALUES
.


THE ALGEBRAIC
CORRELATING
EQUATION
SHOWN IN THE
PLOT SHOULD
HAVE REPLACED
ALL POWER
-
LAW
EXPRESSIONS,
BUT IT DID NOT…


WHY?



BECAUSE IT WAS
NOT
REPRODUCED IN
MOST OF THE
POPULAR
TEXTBOOKS.


THE FRACTION OF THE LOCAL HEAT FLUX DENSITY DUE TO THE TURBULENT
EDDIES


IN 2000, THE MODEL OF CHAN AND CHURCHILL FOR FLOW WAS EXTENDED TO CONVECTION
USING: .


IT PROVES CONVENIENT TO REPLACE IN THE DIFFERENTIAL ENERGY BALANCE BY A
MORE CONSTRAINED VARIABLE, NAMELY THE TURBULENT PRANDTL NUMBER RATIO:
.



PETER ABBRECHT IN 1956

DETERMINED THE EDDY CONDUCTIVITY EXPERIMENTALLY IN A
DEVELOPING TEMPERATURE FIELD AND CONFIRMED HIS CONJECTURE THAT THIS RATIO IS
INDEPENDENT OF THE TEMPERATURE FIELD AND THEREBY OF THE THERMAL BOUNDARY
CONDITION. IT FOLLOWS THAT , AND
Pr
t

/Pr
ARE, AS WELL.


FROM THE ANALOGY OF MACLEOD IT FOLLOWS THAT
k
t
/
k
, , AND
Pr
t
/Pr
,

ARE
IDENTICAL FOR A ROUND TUBE AND A PARALLEL
-
PLATE CHANNEL IN TERMS OF
a
+

AND
b
+
,
RESPECTIVELY.







j
v
T
c
v
T
j
T
T
k
T
p
w
w
w
'
'
'
'

and

2
/
1










]
)
'
'
(
1
[
)
'
'
(
]
)
'
'
(
1
[
)
'
'
(
Pr
Pr











v
u
v
T
v
T
v
u
t


)
'
'
(
v
T


)
'
'
(
v
T


)
'
'
(
v
T
ALGEBRAIC ANALOGIES BETWEEN MOMENTUM AND HEAT TRANSFER

THE REYNOLDS ANALOGY


OSBORNE REYNOLDS IN 1874 POSTULATED THAT MOMENTUM AND ENERGY WERE
TRANSPORTED AT EQUAL MASS RATES FROM THE BULK OF THE FLUID TO THE WALL BY THE
OSCILLATORY RADIAL MOTION OF TURBULENT EDDIES AND THEREBY OBTAINED A RESULT
THAT CAN BE EXPRESSED IN MODERN TERMS FOR A ROUND TUBE AS:






Nu =Pr Re
(
f/2
)


THE PRANDTL
-
TAYLOR ANALOGY



PRANDTL G.I. TAYLOR INDEPENDENTLY IN 1910 AND 1916 DEVISED AN IMPROVEMENT,
NAMELY:









THE REICHARDT ANALOGY



THE ANALOGY DEVELOPED BY REICHARDT IN 1951 IS FAR MORE ACCURATE FUNCTIONALLY
AND NUMERICALLY THAN THAT OF PRANDTL AND TAYLOR, BUT IS ALSO FAR MORE
COMPLICATED




ITS BASIC STRUCTURE HAS BEEN UTILIZED IN MOST SUBSEQUENT ANALOGIES, INCLUDING
THOSE OF FRIEND & METZNER IN 1958, PETUKHOV IN 1970, GNIELINSKI IN 1976, AND MY
OWN IN 1997

2
/
1
)
2
/
)(
1
(Pr
1
2
/
Re
Pr
f
f
Nu
s






CHURCHILL, SHINODA, AND ARAI IN 2000 NOTED THAT THE REICHARDT ANALOGY COULD BE
INTERPRETED AS AN INTERPOLATING EQUATION IN THE FORM OF THE
CUE.


CHURCHILL AND ZAJIC IN 2001 TOOK ADVANTAGE OF THIS REINTERPRETATION TO DEVISE
GREATLY IMPROVED ANALOGIES FOR ALL VALUES OF
Pr

AND
Re
. AS AN EXAMPLE, THEIR
EXPRESSION FOR
Pr ≥ Pr
t

IS:







.



























Nu
Nu
Nu
t
t
1
Pr
Pr
1
1
Pr
Pr
1
3
/
2
1
2
/
1
3
/
1
1
2
Re
Pr
Pr
07343
.
0

and

2
/
Pr
Re


Here

















f
Nu
f
Nu
t
A REINTERPRETATION AND IMPROVEMENT OF THE REICHARDT
ANALOGY



THE COLBURN ANALOGY


ALIAN COLBURN IN 1933 COMBINED THE FOLLOWING EMPIRICAL CORRELATING
EQUATIONS OF E.C. KOO, A DOCTORAL STUDENT AT MIT, FOR THE FRICTION FACTOR, AND
OF DITTUS AND BOELTER FOR THE NUSSELT NUMBER:



f =
0.046/
Re
.0.2


AND


Nu = ARe
.0.8
Pr
n
.


HE TOOK THE RATIO OF THESE TWO EXPRESSIONS, CHOSE AN ARBITRARY VALUE OF
A

=
0.023, AND A ROUNDED
-
OFF VALUE FOR
n
TO OBTAIN:






f/2 = Nu/RePr
1/3
.


HE NAMED THE GROUPING ON THE RIGHT
-
HAND SIDE THE
j


FACTOR.


THIS EXPRESSION TOGETHER WITH AN EMPIRICAL CORRELATING EQUATION FOR THE
FRICTION FACTOR, REMAINS IN USE TO THIS DAY, ALTHOUGH, AS I WILL SHOW YOU, IT IS
SERIOUSLY WRONG FUNCTIONALLY IN EVERY RESPECT AND NUMERICALLY AS WELL.

A GRAPHICAL COMPARISON OF THE ACCURACY OF THE PREDICTIONS OF
SEVERAL ANALOGIES


OUR NEW ANALOGY SHOULD REPLACE ALL PRIOR ANALOGIES AND
CORRELATING EQUATIONS BECAUSE IT IS BOTH SIMPLER AND MORE
ACCURATE.

DEVISING ALGORITHMS FOR THE NUMERICAL SOLUTION OF THE EQUATIONS
OF CONSERVATION



THE APPLICATION OF ELECTRONIC COMPUTERS TO TRANSPORT, BEGINNING AROUND 1950,
ORIGINALLY REQUIRED THE DEVELOPMENT OF SPECIAL
-
CASE ALGORITHMS. THIS WAS AN
IMPORTANT ELEMENT IN THE WORK OF MY DOCTORAL STUDENTS, AS OUTLINED HERE.


NATURAL CONVECTION IN ENCLOSURES


THE FIRST NUMERICAL SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATIONS OF
CONSERVATION BY WILLIAM R. MARTINI IN 1952. HIS SOLUTION WAS INCOMPLETE BUT
DEMONSTRATED PROMISE.


THE FIRST COMPLETE TWO
-
DIMENSIONAL NUMERICAL SOLUTION OF THE PARTIAL
DIFFERENTIAL EQUATIONS OF CONSERVATION BY J. DAVID HELLUMS IN 1960.


THE USE OF A STREAM
-
FUNCTION AND VORTICITY FORMULATION FOR NUMERICAL
SOLUTIONS BY JAMES O. WILKES IN 1963


THE CONCEPT OF A FALSE TRANSIENT FOR THE STREAM
-
FUNCTION FOR NUMERICAL
SOLUTIONS BY M. R. SAMUELS IN 1967


THE CONCEPT OF THE VECTOR POTENTIAL AND THE FIRST THREE
-
DIMENSIONAL NUMERICAL
SOLUTIONS BY KHALID AZIZ, GEORGE HIRASAKI, AND DAVID HELLUMS AT RICE UNIVERSITY
IN 1967


A NON
-
CONSERVATIVE FORMULATION TO IMPROVE CONVERGENCE BY HUMBERT H.
-
S. CHU
IN 1976


THE DYNAMIC DISPLAY OF COMPUTED STREAKLINES BY PAUL P.
-
K. CHAO IN 1982



THE DISCOVERY THAT OSCILLATIONS IN RAYLEIGH
-
BÉNARD
-
TYPE CONVECTION ARE
BETWEEN PLANFORMS BY HIROYUKI OZOE AND COWORKERS AT KYUSHU UNIVERSITY


THE USE OF PHOTOGRAPHED PARTICLE STREAKLINES AND COMPUTED ONES TO DISPLAY
THREE
-
DIMENSIONAL MOTION BY HIROYUKI OZOE AND CO
-
WORKERS AT OKAYAMA
UNIVERSITY IN 1983


OTHER RELATED APPLICATIONS INVOLVING THE DESIGN OF NUMERICAL ALGORITHMS



A TRANSIENT SOLUTION FOR STEADY STATE CONCENTRIC FLOW BY WARREN SEIDER IN 1971



THE USE OF THE MARKER
-
AND
-
CELL METHOD TO LOCATE THE MOVING BOUNDARY IN
THREE
-
DIMENSIONAL EXTRUSION BY EDDY A. HAZBUN IN 1973


THE DISCOVERY OF OSCILLATIONS IN CZOCHRALSKI CRYSTALLIZATION BY VICKI BOOKER AND
COWORKERS AT TSUKUBA UNIVERSITY IN 1995


DEFINITIVE STUDIES OF COMBINED MAGNETIC AND GRAVITATIONAL CONVECTION BY
HIROYUKI OZOE AND COWORKERS AT KYUSHU UNIVERSITY AND AGH UNIVERSITY, KRAKOW


HANRATTY AND CO
-
WORKERS DEVISED, BEGINNING IN 1995, LAGRANGIAN ALGORITHMS
FOR DNS CALCULATIONS, AND THEREBY CONFIRMED THEORETICALLY THEIR 1977
OBSERVATION, BASED ON ELECTROCHEMICAL MEASUREMENTS, THAT THE DEPENDENCE OF
MASS TRANSFER ON Sc DIFFERS FROM THAT OF HEAT TRANSFER ON Pr.


THE LESSON FROM THESE EXAMPLES IS THAT THE PREDICTION OR SIMULATION OF TRANSPORT
OFTEN DEPENDS ON CONCEPTUAL INNOVATION, EITHER MATHEMATICALLY OR PHYSICALLY

SIMULATION


SIMULATION IN THE CURRENT SENSE ALLOWS US TO USE CORRELATIONS FOR TRANSPORT
TO PREDICT COMPLEX BEHAVIOR FOR THE PURPOSES OF DESIGN AND ANALYSIS.


THE ADVANCEMENT AND CURRENT STATE OF SIMULATION ARE BEYOND THE SCOPE OF MY
PRESENTATION. HOWEVER, IT IS APPROPRIATE HERE TO NOTE THAT THIS PROCESS INVOKES
A HIDDEN RISK, NAMELY THE POSSIBLE ERROR DUE TO OUT
-
OF
-
DATE AND ERRONEOUS
CORRELATING EQUATIONS IMBEDDED IN COMPUTER PACKAGES.


SIMULATION HAS ANOTHER ROLE THAT HAS BEEN IMPLICIT IN THIS PRESENTATION, NAMELY
THE PREDICTION OF DETAILED BEHAVIOR FROM “FIRST PRINCIPLES” IN ORDER TO PRODUCE
“COMPUTED VALUES” AS A SUPPLEMENT TO EXPERIMENTAL DATA IN THE CONSTRUCTION
OF CORRELATING EQUATIONS.

SUMMARY


I HAVE PRESENTED A FEW ILLUSTRATIONS OF OUR PROGRESS OVER THE PAST CENTURY IN
PREDICTING TRANSPORT. THEY ARE JUST THAT


ILLUSTRATIONS; I HAVE BEEN HIGHLY
ARBITRARY IN MY CHOICES.



SOME OF THE PROGRESS CONSISTS OF THE ABANDONMENT OF FAMILIAR CONCEPTS


A
PAINFUL PROCESS AND ONE THAT RISKS THE APPEARANCE OF CRITICISM OF IDOLS OF MINE
AS WELL AS YOURS. I TRUST THAT IF THEY WERE HERE THEY WOULD APPROVE. IN THAT
REGARD, I RECALL W.K. LEWIS, IN AN ANECDOTAL LECTURE AT AN AIChE MEETING,
MENTIONING THAT THE
LEWIS NUMBER

COMMEMORATED HIS WORST CONCEPTUAL
ERROR.



MOST OF THE ADVANCES THAT I HAVE DESCRIBED TODAY ORIGINATED IN ACADEMIC
RESEARCH. I WAS ABLE TO IDENTIFY ADVANCES STEMMING FROM INDUSTRIAL RESEARCH
ONLY IN THE RARE INSTANCES WHEN THEY HAVE BEEN RELEASED FROM SECRECY AND
APPEARED IN THE LITERATURE.



THE ULTIMATE CERTIFICATION OF ADVANCES IN TRANSPORT IS THEIR ADOPTION FOR
DESIGN, OPERATION, AND ANALYSIS, BUT THAT IS DIFFICULT TO QUANTIFY, EXCEPT
PERHAPS BY THEIR “APPEARANCE” IN COMPUTATIONAL PACKAGES. A FEW ADVANCES WERE
MENTIONED THAT HAVE A MORE LIMITED BUT NEVERTHELESS IMPORTANT ROLE, NAMELY
IMPROVEMENT IN UNDERSTANDING.


CONCLUSIONS


SOME OF YOU WHO WORK IN PROCESS DESIGN OR OPERATION MAY DISMISS WHAT I HAVE
SAID AS MATHEMATICALLY
-
ORIENTED AND ONCE
-
REMOVED FROM PRACTICALITY. THAT IS A
DANGEROUS INFERENCE.



IN THE EARLY DAYS OF THE AIChE, CORRELATING EQUATIONS WERE DEVISED BY DRAWING A
STRAIGHT LINE THROUGH A LOG
-
LOG PLOT OF EXPERIMENTAL DATA. OVER THE CENTURY
WE HAVE COME TO REALIZE THAT EXPRESSIONS SO
-
DERIVED ARE ALMOST CERTAINLY IN
ERROR FUNCTIONALLY, AND THEREBY MAY BE IN SERIOUS ERROR NUMERICALLY AS WELL,
OUTSIDE OF A NARROW RANGE. IF YOU ARE CLINGING TO ANY SUCH EXPRESSIONS
INVOLVING PRODUCTS OF ARBITRARY POWER
-
FUNCTIONS, FOR EXAMPLE
THE COLBURN
ANALOGY
, YOU ARE DANGEROUSLY OUT OF DATE.



OVER THE CENTURY, THE MOST OBVIOUS CHANGE IN PREDICTING TRANSPORT IS THE
DEVELOPMENT OF POWERFUL COMPUTER HARDWARE AND USER
-
FRIENDLY SOFTWARE.
HOWEVER, THE ACCURACY OF NUMERICAL SOLUTIONS DEPENDS CRITICALLY UPON THE
VALIDITY OF THE MODEL, ON CONVERGENCE, AND ON STABILITY.



TRUSTING MODELS AND/OR THEIR SOLUTIONS, WHOSE LIMITS OF ACCURACY AND
VALIDITY HAVE NOT BEEN TESTED WITH EXPERIMENTAL DATA, IS EQUIVALENT TO BELIEVING
IN THE EASTER BUNNY.



THE MOST RELIABLE EXPRESSIONS FOR THE PREDICTION OF TRANSPORT ARE THOSE THAT
HAVE A THEORETICAL STRUCTURE AND HAVE BEEN CONFIRMED BY BOTH EXPERIMENTAL
DATA AND NUMERICAL SIMULATIONS. THE PRINCIPAL IMPROVEMENT NEEDED WITH
RESPECT TO THE PREDICTION OF TRANSPORT IS FOR A METHODOLOGY FOR THE ACCURATE
PREDICTION OF
DEVELOPING

TURBULENT FLOW AND OF CONVECTION IN THAT REGIME.
THE
k
-
ε
MODEL PURPORTS TO FULFILL THIS NEED, BUT IT IS HIGHLY INACCURATE NEAR THE
WALL, WHICH IS THE MOST CRITICAL REGION.



THE CURRENT LACK OF INTEREST IN AND SUPPORT FOR RESEARCH IN TRANSPORT IS
PRIMARILY A CONSEQUENCE OF THE SHIFT OF SUPPORT AND INTEREST FROM CHEMICAL TO
BIOLOGICAL PROCESSING, AND OF A RELATED SHIFT OF INTEREST FROM PROCESS TO
PRODUCT DESIGN. I FORESEE A PARTIAL REVERSAL AS ENERGY CONVERSION BECOMES A
NATIONAL FOCUS.



SOONER OR LATER IT WILL BE REALIZED THAT A FUNDAMENTAL AND
BROAD UNDERSTANDING OF TRANSPORT IS ESSENTIAL FOR IMPROVED
PROCESSING; WHETHER THERMAL, CHEMICAL, OR BIOLOGICAL,
WHETHER BATCH OR CONTINUOUS, AND WHETHER ON A NANO
-
SCALE
OR A MACRO
-
SCALE. ITS STUDY AND ADVANCEMENT REMAIN ESSENTIAL
TO CHEMICAL ENGINEERING.