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ORIGINS OF DIMENSIONLESS GROUPS OF

HEAT AND MASS TRANSFER

PREPARED BY

U. GRIGULL, H. SANDNER, J. STRAUB, H. WINKLER

ON THE OCCASION OF THE 7TH INTERNATIONAL HEAT TRANSFERCONFERENCE, MUNCHEN 1982

LEHRSTUHL A FUER THERMODYNAMIK TECHNISCHE UNIVERSITAETMUENCHEN

BARON JEAN BAPTISTE JOSEPH

FOURIER

1768-1830

French mathematician and physicist famous for hispioneer work on the representation of functions bytrigonometric series, was born at Auxere on March21, 1768, the son of a tailor. He became a teacher inmathematics in 1784 at the military school there. Hetaught at the Ecole Normale at Paris from itsfounding in 1795, where his success soon led to theoffer of the chair of analysis at the EcolePolytechnique. In 1807 he was made a member of theacademy of sciences.

Fourier's masterpiece was his mathematical theory ofheat conduction stated in "Theorie Analytique de laChaleur" (1822), one of the most important bookspublished in the 19th century. It marked an epochboth in the history of pure and of appliedmathematics, for in it Fourier developed the theory of

the series known by his name and applied it to thesolution of boundary-value problems in partialdifferential equations. This work brought to a close along controversy, and henceforth it was generallyagreed that almost any function of a real variable canbe represented by a series involving the sines andcosines of integral multiples of the variable. Fourier

ORIGINS OF DIMENSIONLESS GROUPS OF HE... file:///home/teja/Desktop/ORIGINS OF DIMEN

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died in Paris on May 16, 1830.

JEAN BAPTISTE BIOT

1774-1862

French physicist, best known for his work inpolarization of light, was born in Paris on April 21,1774. In 1800 he became professor of phsysics at theCollege de France, through the influence of Laplace,from whom he had sought and obtained the favour ofreading the proof sheets of the "Mecanique Celeste".

J. B. Biot, although younger than Fourier, worked onthe analysis of heat conduction even earlier - in 1802

or 1803. He attempted, unsuccessfully, to deal withthe problem of incorporating external convectioneffects in heat conduction analysis in 1804. Fourierread Biot's work and by 1807 had determined how tosolve the problem.

In 1804 he accompanied Gay Lusac on the firstballoon ascent undertaken for scientific purposes, in1820, with Felix Savart, he discovered the law knownas "Biot and Savart's Law". He was especiallyinterested in questions relating to the polarization of

light, and for his achievements in this field he wasawarded the Rumford Medal of the Royal Society in1840. He died in Paris on February 3, 1862.


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JEAN CLAUDE EUGENE PECLET

1793-1857

French physicist, born February 10, 1793 atBesancon, became one of the first scholars of the

Ecole Normale at Paris, Gay Lussac and Dulongbeing his teachers. Peclet was elected professor atthe College de Marseille in 1816, teaching physicalsciences there until 1827. He returned to Paris whennominated Maitre de Conferences at the EcoleNormale and was elected professor at the importantEcole Centrale des Arts et Manufactures. In 1840 hebecame Inspecteur General de L'Instruction Publiqueand retired from this charge in 1852 to devotehimself exclusively to teaching.

His publications were famous for their clarity of style,sharpminded views and well performed experiments.His famous book "Traite de la Chaleur et de sesApplications aux Arts et Aux Manufactures" (Paris1829) was distributed world-wide and had beentranslated in German.

Peclet continued lecturing until his death December6, 1857 at Paris.

FRANZ GRASHOF

1826-1893

German engineer, born July 11, 1826 at Dsseldorf,left school at the age of 15 to work as a mechanic,and then attended trade school in Hage andsecondary school in Dsseldorf. From 1844 until1847 Grashof studied mathematics, physics and

machine design at the Berlin Royal TechnicalInstitute. After a voyage of nearly three years whichtook him as far as the Dutch Indies and Australia hecontinued his studiesat Berlin in 1852.

Grashof was one of the leaders in founding theSociety of German Engineers (Verein Deutscher




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Ingenieure, VDI) and assumed an enormous load asauthor, editor, corrector and dispatcher. In 1863Redtenbacher died and Grashof's name was soesteemed that the Technical University of Karlsruheappointed him to be successor as superindependentof the engineering school. He also served as professor

of applied mechanics and mechanical engineeringand his lectures included strength of materials,hydraulics and theory of heat, in addition to generalengineering.

After Grashof's death, October 26, 1893 at Karlsruhethe Society of German Engineers honored his memoryby the institution of the Grashof CommemorativeMedal as the highest distinction that the societycould bestow for merit in the engineering skills.

JOSEF STEFAN

1835-1893

Austrian physicist, whose original contributionsranged over several important fields, including thekinetic theory of gases, hydrodynamics and inparticular, radiation, was born on March 24, 1835 atSt. Peter near Klagenfurt and died on January 7,1893 in Wien.

Stefan was educated at the University of Wien,receiving his doctor of philosophy in 1858, thenbecame privatdozent in mathematical physics, in1863 professor ordinarius of physics and in 1866director of the physical institute. He was adistinguished member of the Academy of Sciences ofWien, of which he was appointed secretary in 1875.Before Stefan's work, G. R. Kirchhoff had already

described the perfect radiator as the "Perfect BlackBody", namely, one that absorbed all the radiationthat fell on it and reflected none, but emittedradiation of all wave lengths. Stefan showedempirically in 1879 that the radiation of such a bodywas proportional to the fourth power of its absolutetemperatures, a relationship known as the "Stefan &




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Boltzmann" law after it had been deduced by L.Boltzmann in 1884 from thermodynamicconsiderations.

In the year 1891 Stefan published his work on theformation of ice in the polar seas, giving a special

solution of this nonlinear conduction problem withphase change (the more general solution being dueto F. Neumann).

ERNST MACH

1838-1916

Austrian physicist and philosopher, whose work, both

in physics and in philosophy, had a great influence on20th-century thought, was born on February 18,1838, at Turas in Moravia and educated in Wien. Hewas professor of physics at Graz from 1864 to 1867and at Prag from from 1867 to 1895, and professor ofinductive philosophy at Wien from 1895 to 1901. Hewas made a member of the Austrian House of Peers in1901 and died at Mnchen on February 19, 1916.

Mach was a throughgoing positivist and took theview, which most scientists now share, that no

statement is admissible in natural science unless it isempirically verifiable. His criteria of verifiabilitywere, however exceptionally rigorous: They led himnot only to reject such metephysical conceptions butalso to oppose the introduction of atoms andmolecules into the physical theory. Nevertheless itwas his criticism along the lines of Sir Isaac Newton'ssystem that made the way clear for Albert Einstein'stheory of relativity. As a positivist, he regardedscientific laws as purely descriptive; and he held that

the choice between the alternative hypothesescovering the same facts was to be made on thegrounds of economy.

Mach's name is associated with the Mach Number,which expresses the speed of matter relative to thelocal speed of sound.




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OSBOURNE REYNOLDS

1842-1912

English engineer and physicist, best known for his

work in the field of hydraulics and hydrodynamics,was born at Belfast on Aug. 23, 1842. Gaining earlyworkshop experience and graduating at Queens'College Cambridge in 1867, he became the firstprofessor of engineering in the Owens College,Manchester in 1868. He was elected a fellow of theRoyal Society in 1877 and a Royal Medallist in 1888.

Reynolds' studies of condensation and the transfer ofheat between solids and fluids brought radical

revision in boiler and condenser design, while hiswork on turbine pumps laid the foundation of theirrapid development. A fundementalist amongengineers, he formulated the theory of lubrication(1886), and in his classical paper on the law ofresistance in parallel channels (1883) investigatedthe transition from smooth, or laminar, to turbulentflow, later (1889) developing the mathematicalframework which became standard in turbulencework.

His name is perpetuated in the "Reynolds Number",which provides a criterion for dynamic similarity andhence for correct modelling in many fluid flowexperiments.

Among his other work was teh explanation of theradiometer and an early absolute determination ofthe mechanical equivalent of heat. Reynolds retiredin 1905 and died at Watchet, Somerset, on Feb. 21,1912.




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LORD RAYLEIGH

1842-1919

British physicist, who was awarded the NobelPrize for physics in 1904 for his discovery

(1894) of the inert elementary gas Argon, incollaboration with Sir William Ramsay, was bornnear Maldon, Essex, on Nov. 12, 1842, andeducated at Trinity College, Cambridge, wherehe graduated senior Wrangler (1865). Assuccessor to James Clerk Maxwell he was headof the Cavendish Laboraty at Cambridge from1879 to 1884, and in 1887 he became professorof Natural Philosophy in the Royal Institution ofGreat Britain. Elected (1873) a fellow of the

Royal Society, he was president from 1905 to1908.

His research almost covered the entire field ofphysics, including sound, wave theory, optics,colour vision, electrodynamics,electromagnetism, the scattering of light,hydrodynamics, the flow of liquids, capillary,viscosity, the density of gases, photgraphy andelasticity, as well as electrical measurementsand standards. His research on sound wasembodied in his "Theory of Sound", and hisother extensive studies in physics appeared inhis "Scientific Papers". Rayleigh died on June30, 1919 at Witham, Essex.




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LEO GRAETZ

1856-1941

German physicist, born at Breslau on September26, 1856, studied mathematics and physics at

Breslau, Berlin and Strassburg. In 1881 hebecame assistant to A. Kundt at Strassburg and in1883 he went to the University of Mnchen,where he became a professor in 1908 andoccupied the Second Chair for physics parallel toroentgen.

His scientific work was first concerned with thefields of heat conduction, radiation, friction andelasticity and after 1890 with problems of

electromagnetic waves and cathode rays.Graetz was a profilic technical writer, with 23editions of his "Electricity and its Applications"and a five volume work "Handbook of Electricityand Magnetism", which contributed to the widedissemination of knowledge in electricity, which atthat time was in its infancy. He died in Mnchenon November 12, 1941.

SIR THOMAS EDWARD STANTON

1865-1931

British engineer, born at Atherstone in Warwickshireon December 12, 1865. In 1888 he entered OwensCollege, Manchester, and followed the engineeringcourse in the Whitworth Laboratory under OsborneReynolds. After taking the degree of B.Sc. in 1891 atthe Victoria University, with First-Class Honours, he

continued to work in Reynolds' laboratory, at first asjunior and later as senior demonstrater, until 1896.From 1892 to 1896 he was also resident tutor inmathematics and engineering at the Hulme Hall ofResidence, Manchester. In June, 1896, Stantonobtained a post as senior assistant lecturer inengineering at University College, Liverpool, under




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professor Hele-Shaw. In December, 1899, he wentfrom Liverpool to Bristol University College asprofessor engineering.

In July, 1901, he was offered the position ofsuperintendent of the Engineering Department of the

National Physical Laboratory in Bristol. This post hecontinued to hold until his retirement from officialduties in December, 1930, at the age of 65, one yearbefore his death.

Stanton's main field of interest was fluid flow andfriction, and the related problem of heattransmission, from 1902 to 1907 he excuted a largeresearch program concerning wind forces onstructures, such as bridges and roofs. After 1908, theyear when the Wright Brothers made their firstaeroplane flights in Europe, Stanton devoted toproblems of aeroplane and airship design and thedissipation of heat from air-cooled engines

LUDWIG PRANDTL

1875-1953

German physicist famous for his work in aeronautics,

was born at Freising, Bavaria, on February 4, 1875.He qualified at Mnchen in 1900 with a thesis onelastic stability, and was professor of appliedmechanics at Gttingen from 1904 until his deaththere on August 15, 1953. In 1925 he becamedirector of The Kaiser Wilhelm Institute for FluidMechanics. His discovery (1904) of the "BoundaryLayer" which adjoins the surface of a body moving ina fluid led to an understanding of skin friction dragand of the way in which streamliningreduces the

drag of airplane wings and other moving bodies. Hiswork on wing theory, published in 1918-1919, whichfollowed that of F.W. Lanchester (1902-1907) but wascarried out independently. elucidated the flow overplane wings of finite span.

Prandtl made decisive advances in boundary layer




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and wing theories, and his work became the basicmaterial of aeronautics. He also made importantcontributions to the theories of supersonic flow and ofturbulence, besides contributing much to thedevelopment of wind tunnels and other aerodynamicequipment. In addition, he devised the soap-film

analogy for the torsion of noncircular sections andwrote on the theory of plasticity and of meteorology.

MAX JAKOB

1879-1955

German physicist, born July 20, 1879 inLudwigshafen, studied electrical engineering at the

Technical University of Mnchen where hegraduated in 1903. From 1903 until 1906 he was anassistant to O. Knoblauch at the Laboratory forTechnical Physics. After working in the electricalindustry Jakob joined the "Physikalisch-TechnischeReichsanstalt" at Berlin-Charlottenburg in 1910,where he started his career in thermodynamics andheat transfer. He conducted a large amount ofimportant works in these fields, covering such areasas steam and air at high pressure, devices for

measuring thermal conductivity, the mechanisms ofboiling and condensation, flow in pipes and nozzles,and others.

In 1936 he emigrated to the United States, where hebecame a research professor at the Illinois Instituteof Technology and consultant in Heat Transfer for theArmour Research Foundation. In 1952, three yearsbefore his sudden death, he was awarded theWorchester Reed Warner Medal by the AmericanSociety of Mechanical Engineers.

His long years of research resulted in significantcontributions to the literature of the profession;nearly 500 books, articles, reviews and discussionshave been published.




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WILHELM NUSSELT

1882-1957

German engineer, born November 25, 1882 atNrnberg, studied machinery at the Technical

Universities of Berlin-Charlottenburg and Mnchenwhere he graduated in 1904 and conductedadvanced studies in mathematics and physics. Hebecame an assistant to O. Knoblauch at theLaboratory for Technical Physics in Mnchen andcompleted his doctoral thesis on the conductivity ofinsulating materials in 1907, using the "NusseltSphere" for his experiments. From 1907 to 1909 heworked as an assistant of Mollier in Dresden,qualifying himself for a professorship with a work on

heat and momentum transfer in tubes.

In 1915 Nusselt published his pioneer paper: "TheBasic Laws of Heat Transfer" in which he firstproposed the dimensionless groups now known as theprincipal parameters in the similarity theory of heattransfer. Other famous works were concerned withthe film condensation of steam on vertical surfaces,the combustion of pulverized coal and the analogybetween heat and mass transfer in evaporation.Among the primarily mathematical works of Nusselt,the well known solutions for laminar heat transfer inthe entrance region of tubes, for heat exchange incross-flow and the basic theory of regeneratorsshould be mentioned.

Nusselt was professor at the Technical Universities ofKarlsruhe from 1920 to 1925 and at Mnchen from1925 until his retirement in 1952. He was awardedthe Gauss-Medal and the Grashof CommemorativeMedal. Nusselt died in Mnchen on September1,

1957.




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ERNST SCHMIDT

1892-1975

German scientist and pioneer in the field ofengineering thermodynamics, especially in heat and

mass transfer, was born on February 11, 1892 atVgelsen, near Lneburg. He studied civil andelectrical engineering at Dresden and Mnchen, andjoined the Laboratory for Applied Physics at theTchnical University, Mnchen, in 1919, which wasthen under the direction of Oscar Knoblauch. One ofhis early research efforts there was a carefulmeasurement of the radiation properties of solids,which caused him to propose and develop the use ofaluminium foil as an effective radiation shield.

In 1925 he received a call to come as professor anddirector of the Engineering Laboratory to theTechnical University, Danzig. Here he publishedpapers on the now well known graphical differencemethod for unsteady heat conduction, and on theSchlieren and Shadow method to make thermalboundaries visible and to obtain local heat transfercoefficients. He was the first to measure the velocityand temperature field in a free convection boundarylayer and the large heat transfer coefficientsoccuring in droplet condensaton. A paper pointingout the analogy between heat and mass transfercaused the dimensionless quantity involved to becalled "Schmidt Number".

In 1937 he became director of the Institute forPropulsion of the newly founded AeronauticalResearch Establishment at Braunschweig andprofessor at the University there. In 1952 Schmidtoccupied the chair for thermodynamics at the

Technical University of Mnchen which before himhad been held by Nusselt. Being strongly involved inthe development of the international steam tables,Schmidt continued his scientific activity after hisretirement (1961) until his death in 1975.

In recognition of his work, he had received the




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Ludwig Prandtl Ring, The Max Jacob Award and TheGrashof Commemorative Medal.

THOMAS KILGORE SHERWOOD

1903-1976

Sherwood, born in Columbus, Ohio, July 25, 1903,was one of America's great chemical engineers; hisenergy, research contributions, applied engineeringachievements, and influence on chemical engineeringeducation were prodigious.

Sherwood came to M.I.T. in 1923 for graduate workin the Chemical Engineering Department and

completed his doctoral thesis under Warren K. Lewis.entitled "The Mechanism of the Drying of Solids" in1929. From 1930 to 1969 he was professor at M.I.T.contributing decisively to the standards of excellenceof this famous institution.

Sherwood's primary research area was mass transferand its interaction with flow and with chemicalreaction and industrial process operations in whichthose phenomena played an important part. His rapidrise to the position of world authority in the field of

mass transfer was accelerated by the appearence ofhis book, "Absorption and Extraction", the firstsignificant text in this area, published in 1937.Completely rewritten, with Pigford and Wilke in 1974under the title "Mass Transfer", the book has hadenormous influence. The worldwide use of theSherwood Number is a memorial to that effort

In addition to three honorary doctorates manyawards were bestowed on Sherwood, such as the U.S.

Medal for Merit in 1948 and the Lewis Award in1972. He died on January 14, 1976.




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ALLAN PHILIP COLBURN

1904-1955

American Engineer, born in Madison, Wisconsin, onjune 8, 1904 graduated from the University of

Wisconsin in Chemical Engineering with highhonors in 1926 and was awarded an engineeringfellowship for graduate studies. He received hisMaster of Science in 1927 and his Ph.D. in 1929.

His research was on condensation of water vaporfrom saturated air streams, a topic that in itsbroader aspects interested him to the end of hislife. He brought together for the first time inAmerican engineering work the fundamentals of

momentum, heat and mass transfer along withthermodynamic principles to deal with this complexproblem.

Although not known formally, perhaps, as adimensionless parameter, the empirical ColburnJ-Factor is indeed an operational one.

Colburn joined the Chemical EngineeringDepartment at the University of Delaware in 1938.He was appointed as assistant to the President of

the University in 1947, Acting President in 1950and Provost and Coordinator of Scientific Researchuntil his death in 1955.




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