A CENTURY OF TRANSPORT A Personal Tour by Stuart W. Churchill DEPARTMENT OF CHEMICAL AND BIOMOLECULAR ENGINEERING THE UN

1. A CENTURY OF TRANSPORT A Personal Tour by Stuart W. Churchill DEPARTMENT OF CHEMICAL AND BIOMOLECULAR ENGINEERING THE UNIVERSITY OF PENNSYLVANIA

2. 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

3. 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.

4. 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:

5. “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.”

6. 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 :

7. 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

8. 12) POTENTIAL FLOW AND THE THIN-BOUNDARY-LAYER CONCEPT 13) FREE STREAMLINES PREDICT = 0.611 FOR ORIFICE. THE COEFFICIENTREAL VALUE IS0.5793. 14) CRITERIA FOR TURBULENT FLOW IN PIPES: OSBORNE REYNOLDS IN 1883 REYNOLDS: Re = 2100OR a+= a(τw ρ)½/μ = Re(f/8)½ 56 MODERN: LAMINAR: a+≤45 [Re ≤ 1600]; TURBULENT: a+ ≥ 150 [Re ≥ 4020]

9. 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)/ρ BYgβ(T – T∞) 17) THE RADIATIVE HEAT TRANSFER COEFFICIENT LINEARIZATION ALLOWS USE WITH OHM’S LAWS 18) BLACK-BODY AND GRAY-BODY RADIATION

10. 19) ASYMPTOTIC SOLUTIONS FOR TURBULENT FREE CONVECTION NUSSELT IN 1915: h APPROACHES INDEPENDENT FROM x AS x→ ∞ REQUIRES NuxGrx1/3 FRANK-KAMENETSKII IN 1937:hINDEPENDENT OFk AND μ REQUIRESNuxGrx1/2Pr ECKERT AND JACKSONIN 1951: INTEGRAL BOUNDARY LAYER THEORY NuxGrx0.4 CHURCHILL IN 1970: Nux→ A Rax1/3ASPr→ ∞ ANDx→ ∞ Nux→B (RaxPr)1/3AS Pr→ 0 AND x→ ∞ SEEMINGLYVALIDATED 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

11. 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(ρs− ρ) 2) HEIGHT OF EXPANDED BED:L(1− ε)= L (1− ε) 3) MEAN INTERSTITIAL VELOCITY:um0 = uT εn 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:

12. 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.

13. 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, uTIS 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.

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

16. SIMILARITY TRANSFORMATIONS A FEW FAMILIAR EXAMPLES 1) TRANSIENT THERMAL CONDUCTION 2) THE THIN BOUNDARY-LAYER TRANSFORMATIONOF PRANDTL IN 1904 3) THE POHLHAUSEN TRANSFORMATION OF 1921FOR 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

17. 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:

19. 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 = 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

20. 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 THEALTERNATIVE FORMS:

21. AND • OUR FIRST APPLICATION - LAMINAR FREE CONVECTIONFROM AN ISOTHERMAL VERTICAL PLATE IN THE THIN LAMINAR BOUNDARY LAYER REGIME - RESULTED IN: • GRAPHICAL EVALUATION OF n :

22. FOR n = 9/4, PER THE GRAPH:

23. A SUBSEQUENT EARLY APPLICATION THE VELOCITY DISTRIBUTION IN TURBULENT FLOW IN A ROUND TUBE ASYMPTOTES COMBINATION

24. THE CANONICAL PLOT

25. THE CONVENTIONAL PLOT OF THE SAME VARIABLES

26. 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)

27. 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.

28. MULTIPLE VARIABLES • MAY BE INCORPORATED IN ASYMPTOTES AS IS GrX IN THE PRIOR EXAMPLE, NAMELY: • MAY BE INTRODUCEDSERIALLY, AS IN:

29. 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 ALOWER AND y∞ AN UPPER BOUND, AND VICE VERSA.

30. THIS ANOMALY CAN BE AVOIDED BY USING “STAGGERED” VARIABLES SUCH AS: WHICH FOLLOWS FROMAPPLICATION OF THE CUE TO y0AND y1, NAMELY: AND THEN IN TURN TO y∞, NAMELY:

31. 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

32. 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, BUTTHIS EXPRESSION CAN BE ADAPTED AS A GENERALIZED ONE FOR ALL TRANSITIONAL PROCESSES.

33. THE STATUS AND FUTURE OF THE CUE

34. 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)

35. 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 BYNux , • TO THE LOCAL MIXED-MEAN RATE OF REACTION AS REPRESENTED • BY .

36. ILLUSTRATIVE REPRESENTATIONS • LAMINAR FLOW • HERE K0x = k0x/umIS THE DIMENSIONLESS DISTANCE THROUGH THE REACTOR

37. TURBULENT FLOW

38. 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 1895SPACE-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 DATATHAT f WAS INVERSELY PROPORTIONAL TO Re1/4. • UNFORTUNATELY, THIS IS A CRUDE APPROXIMATION THAT DOES NOT APPLY TO ANY FINITE RANGE OF Re.

39. 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: .

40. 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 LENGTHCONCEIVED 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: .

41. 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 :

42. 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.”

43. 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 OFTHE 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.

44. 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.

45. 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.

46. 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.

47. 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.

48. 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 = 2a+um+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 (POISEUILLE’S LAW), ft = (Re/37530)2, AND fTIS 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.