A CENTURY OF TRANSPORT

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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
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OBJECTIVES

• 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

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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.

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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

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• 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.

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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

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• 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

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• 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(tw
  ?)½ /µ Re(f/8)½ 56
• MODERN LAMINAR a 45 Re 1600
  TURBULENT a 150 Re 4020

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• 15) FULLY-DEVELOPED CONVECTION
• UNIFORM HEATING
• THE NEAR-ATTAINMENT OF ASYMPTOTIC VALUES OF
  (T-T0)/(Tm-T0) AS A FUNCTION OF r/a AND OF
  THE LOCAL HEAT TRANSFER COEFFICIENT
• UNIFORM WALL-TEMPERATURE
• THE NEAR-ATTAINMENT OF ASYMPTOTIC VALUES OF
  (Tw-T)/(Tw-Tm) 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)/? BY gß(T T8)
• 17) THE RADIATIVE HEAT TRANSFER COEFFICIENT
• LINEARIZATION ALLOWS USE WITH OHMS LAWS
• 18) BLACK-BODY AND GRAY-BODY RADIATION

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• 19) ASYMPTOTIC SOLUTIONS FOR TURBULENT FREE
  CONVECTION
• NUSSELT IN 1915 h APPROACHES
  INDEPENDENT FROM x AS x ? 8
• REQUIRES Nux Grx1/3
• FRANK-KAMENETSKII IN 1937 h
  INDEPENDENT OF k AND µ
• REQUIRES Nux Grx1/2 Pr
• ECKERT AND JACKSON IN 1951
  INTEGRAL BOUNDARY LAYER THEORY
• 
  Nux Grx0.4
• CHURCHILL IN 1970 Nux ? A Rax1/3 AS
  Pr ? 8 AND x ? 8
• Nux ? B
  (RaxPr)1/3 AS Pr ? 0 AND x ? 8
  
• SEEMINGLY VALIDATED BY
  LIMITED EXPERIMENTAL DATA
• 20) OHMS DERIVED IN 1827 EXPRESSIONS FOR
  STEADY-STATE ELECTRICAL CONDUCTION REGULARLY
  APPLIED IN CHEMICAL ENGINEERING FOR OTHER LINEAR
  BEHAVIOR

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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- e)g(?s- ?)
• 2) HEIGHT OF EXPANDED BED
  L(1- e) L (1- e)
• 3) MEAN INTERSTITIAL VELOCITY
  um0 uT en
• 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

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• 3) LAMINAR CONDENSATION
• NUSSELT IN 1916 FOR A FILM FALLING DOWN
  A VERTICAL PLATE
• HERE, ? 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.

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• 6) THERMOACOUSTIC CONVECTION
• INCORPORATION OF FOURIERS
  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, uT 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.

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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.
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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

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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

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DIMENSIONAL ANALYSIS OF A LIST OF VARIABLES

• RAYLEIGH HAD THE LAST WORD WHEN IN 1915 HE
  DERIVED
• Nu A Ren Prm B Re2n Pr2m Re3n
  Pr4m .....
• HE EMPHASIZED THAT THIS ONLY MEANT THAT
  
• Nu FRe, 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

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A CORRELATING EQUATION FOR ALMOST EVERYTHING

• IN 1972 WE BEGAN TESTING AS A GENERAL EXPRESSION
  FOR CORRELATION
• WE CALLED THIS THE CHURCHILLUSAGI 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

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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

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FOR n 9/4, PER THE GRAPH
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A SUBSEQUENT EARLY APPLICATION

• THE VELOCITY DISTRIBUTION IN TURBULENT FLOW IN A
  ROUND TUBE
• ASYMPTOTES
• COMBINATION
• 
  
  
  

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THE CANONICAL PLOT
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THE CONVENTIONAL PLOT OF THE SAME VARIABLES
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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)

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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.

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MULTIPLE VARIABLES

• MAY BE INCORPORATED IN ASYMPTOTES AS IS GrX IN
  THE PRIOR EXAMPLE, NAMELY
• MAY BE INTRODUCED SERIALLY, AS IN

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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 y0 IS A LOWER
  AND y8 AN UPPER BOUND, AND VICE VERSA.

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THIS ANOMALY CAN BE AVOIDED BY USING STAGGERED
VARIABLES SUCH AS WHICH FOLLOWS
FROM APPLICATION OF THE CUE TO y0 AND y1 ,
NAMELY AND THEN IN TURN TO y8, NAMELY

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• 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

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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.

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THE STATUS AND FUTURE OF THE CUE
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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)

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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

• THIS EQUATION MAY BE INTERPRETED AS AN
  ANALOGY RELATING
• THE LOCAL RATE OF HEAT TRANSFER, AS
  REPRESENTED BY Nux ,
• TO THE LOCAL MIXED-MEAN RATE OF REACTION AS
  REPRESENTED
• BY .

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ILLUSTRATIVE REPRESENTATIONS

• LAMINAR FLOW
• HERE K0x k0x/um IS THE DIMENSIONLESS DISTANCE
  THROUGH THE REACTOR

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• TURBULENT FLOW

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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 Re1/4.
• UNFORTUNATELY, THIS IS A CRUDE
  APPROXIMATION THAT DOES NOT APPLY TO ANY FINITE
  RANGE OF Re.

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• 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
  .

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• 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 .

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• 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

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• 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.

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• 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 a 0.00700.

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POST-PRANDTL MODELING

• THE k-e MODEL
• FOLLOWS FROM THE CONJECTURES OF KOLMOGOROV,
  PRANDTL, AND BATCHELOR
• EMPIRICAL EQUATIONS FOR k AND e 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.

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• 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-e 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.

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• 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.

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• 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.

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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 PARALLELPLATE
  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
  2aum AND e/a, BUT CONVERGENCE IS VERY RAPID.

49

• THE CORRESPONDING EXPRESSION FOR THE FRICTION
  FACTOR OF ALL REGIMES OF FLOW (LAMINAR,
  TRANSITIONAL, AND TURBULENT) AND ALL EFFECTIVE
  ROUGHNESS RATIOS IS
• HERE, fl 16/Re (POISEUILLES LAW), ft
  (Re/37530)2, AND fT 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

50
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.

51
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.
52
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
  Prt /Pr ARE, AS WELL.
• FROM THE ANALOGY OF MACLEOD IT FOLLOWS THAT kt/k,
  , AND Prt/Pr, ARE IDENTICAL FOR A
  ROUND TUBE AND A PARALLEL-PLATE CHANNEL IN TERMS
  OF a AND b, RESPECTIVELY.

53
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

54
A REINTERPRETATION AND IMPROVEMENT OF THE
REICHARDT ANALOGY

• 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 Prt IS
• 
  
  .

55

• 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.0.8Prn.
• 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/RePr1/3 .
• 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.

56
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.
57
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

58

• 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

59
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

60
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.

61
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.

62
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.

63

• 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-e
  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.

64

• 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.