Come calcolare il numero di Galileo (Ga) rapporto adimensionale che misura le forze gravitazionali e di galleggiamento rispetto alle forze viscose al quadrato in problemi di sedimentazione + caduta libera + letto fluidizzato + powder technology: Ga = g * L^3 / nu^2. g gravita 9.81 m/s^2 Earth; L lunghezza caratteristica m (diametro particella sferica + sphere ball drop); nu viscosità cinematica m^2/s del fluido (acqua 25°C 1e-6; aria 25°C 1.5e-5; oli 1e-4-1e-3; mercurio 1.1e-7).

Numero di Archimede: Ar = Ga * (rho_p - rho_f)/rho_f + correzione per densità relativa particella/fluido.

Reynolds terminale: Re_t = sqrt(Ga/18) Stokes regime + Re_t = particle Reynolds at terminal velocity v_t.

Classificazione regime sedimentazione: Ga<1 Stokes regime viscous dominant + terminal velocity Stokes formula v_t = (rho_p-rho_f)*g*L^2/(18*nu*rho_f) Stokes 1851 Camb Phil Trans 9:8 + Re_t<0.1 Stokes drag CD=24/Re + clay particles + colloids + microbeads + biological cells; 1<Ga<1000 intermediate transitional drag + Schiller-Naumann 1933 CD=(24/Re)*(1+0.15*Re^0.687) + sand fine + 0.1<Re_t<1000; 1000<Ga<1e5 Newton regime turbulent drag CD~0.44 + sand spheres + gravel + 1000<Re_t<2.5e5; Ga>1e5 high inertia turbulent fully developed + boulders + rocks + Re_t>2.5e5 + crisi resistenza CD ~0.18 dragless + Eiffel sweep.

Drag coefficient regimi: Stokes Re<0.1 CD=24/Re; intermediate 0.1<Re<1000 Schiller-Naumann CD=(24/Re)*(1+0.15*Re^0.687); Newton 1000<Re<2.5e5 CD=0.44 + sphere; crisi resistenza Re>2.5e5 CD~0.18-0.2 + dimpled golf ball reduce.

Letto fluidizzato fluidized bed: U_mf minimum fluidization velocity Wen-Yu 1966 + Ergun equation 1952 + Geldart classification A/B/C/D 1973.

Geldart classification: A aeratable (catalytic FCC) 30-100 micron rho 1500 kg/m^3 + smooth fluidization; B sand-like 100-1000 micron + bubbles; C cohesive <30 micron + powder cement flour + cohesive forces; D spoutable >1000 micron + grains coal.

Numeri correlati: Reynolds Re = rho*v*L/mu; Archimedes Ar = (rho_p-rho_f)/rho_f * Ga; Bond Bo = rho*g*L^2/sigma; Eotvos Eo = Bo; Froude Fr = v/sqrt(g*L); Weber We = rho*v^2*L/sigma; Morton Mo = g*mu^4/(rho*sigma^3). Storia: Galileo Galilei (Pisa Italia 1564 - Arcetri Florence 1642) Universita Pisa medicine 1581 + matematica + Padova Universita 1592-1610 + Firenze Granducato Toscana 1610-1633 fino a processo Inquisizione + casa di Arcetri 1634-1642; 1589-1592 esperimenti caduta libera Torre Pendente di Pisa (legendary, contested by historians, but supported by Stillman Drake) + Galileo dimostra che la velocità di caduta non dipende dalla massa (rifiuta Aristotele) + 'De motu' 1592 unpublished manuscript + 'Discorsi e Dimostrazioni Matematiche intorno a Due Nuove Scienze' 1638 Leiden Elsevier 'Two New Sciences' classic + caduta libera + moto uniformemente accelerato + parabola proiettile + matematica + Galileo Galilei pioneer scientific method + telescope 1609 + Jupiter moons Io Europa Ganymede Callisto + Sun spots + Saturn rings + Venus phases + Galileo Galilei Universal Clock + Galileo Galilei Inquisition 1633 + Tycho Brahe Kepler partner + Salviati friend;

Padova Universita 18 anni 1592-1610 chair geometria meccanica astronomia + Galileo Galilei 'Sidereus Nuncius' 1610 telescope discoveries + 'Dialogo sopra i due massimi sistemi del mondo' 1632 Florence Landini classic dialogue Aristotelean Ptolemaic vs Copernican Salviati Sagredo Simplicio + Inquisition trial 1633 abjuration + house arrest Arcetri 1633-1642 + 'eppur si muove' apocryphal.

George Gabriel Stokes (Skreen Sligo Ireland 1819 - Cambridge UK 1903) Cambridge University Pembroke College + Lucasian Professor Mathematics 1849-1903 (succeeding Joshua King + before P A M Dirac); 1851 paper Trans Cambridge Phil Soc 9:8 'On the effect of the internal friction of fluids on the motion of pendulums' classic Stokes' law drag formula F_d = 6*pi*mu*v*r sphere viscous fluid + Stokes drag + 1845 paper 'On the theories of the internal friction of fluids in motion' Navier-Stokes equations + 1850 fluorescence + 1854 wave theory; Royal Society President 1885-1890 + Smith Prize 1841 + Copley Medal 1893 + classical mathematical physics + Cambridge classic.

L Schiller + A Z Naumann (Germany 1933); 1933 paper Z Verein Deut Ing 77:318 'Uber die grundlegenden Berechnungen bei der Schwerkraftaufbereitung' classical Schiller-Naumann correlation drag intermediate Re.

Sabri Ergun (Turkey 1918-2006) PennState University; 1952 paper Chem Eng Prog 48:89 'Fluid flow through packed columns' classic Ergun equation packed bed pressure drop + viscous Kozeny + inertia Burke-Plummer.

Derek Geldart (UK 1929-) Bradford University + Cambridge; 1973 paper Powder Technol 7:285 'Types of gas fluidization' classical Geldart classification A/B/C/D fluidized bed + powder technology classification.

Chien-Yu Wen + Y H Yu (1966) West Virginia University; 1966 paper AIChE J 12:610 'Mechanics of fluidization' Wen-Yu correlation U_mf minimum fluidization velocity + Ergun extension. Eredita: Crowe-Sommerfeld-Tsuji 'Multiphase Flows with Droplets and Particles' 2nd ed 2011 CRC Press 509 pagine + Kunii-Levenspiel 'Fluidization Engineering' 2nd ed 1991 Butterworth-Heinemann 491 pagine + Yang 'Handbook of Fluidization' 2003 CRC + Davidson-Harrison 'Fluidization' 2nd ed 1971 Academic Press; Clift-Grace-Weber 'Bubbles Drops Particles' 1978 Academic Press classic 380 pagine.

Italia ricerca sedimentazione + fluidized bed + powder technology + multiphase: CNR-IIA Istituto Inquinamento Atmosferico Monterotondo Roma + Sesto Fiorentino 250 particolato atmospheric + sedimentation; CNR-ICMATE Genova + Padova + Lecco 200; CNR-ITAE Messina 200 + fluidized bed combustion biomassa; ENEA Casaccia + Frascati + Brindisi + Trisaia + 11 centri 2700 + biomass fluidized bed gasification + thermal storage; RSE Milano 1000 + fluidized bed combustion CFB; SOGIN SpA Roma 1100 nuclear decommissioning + waste fluidized; OGS Trieste 250 + sedimentation Adriatico marine; CMCC Lecce 200; INFN Italia 5000 + Frascati LNF + L'Aquila + Catania.

Italia università Galileo + sedimentazione + fluidized bed: Politecnico Milano DEIB + DENG + LCM 7500 + Carlo Cercignani Boltzmann legacy 1939-2010 kinetic theory + Renzo Piazza CFD spray + Pier Luigi Bonomi multiphase fluidized bed; Politecnico Torino DENERG + DIATI 5000 + Marco Fiori multiphase + Antonio Tomassetti fluidized bed; Federico II Napoli DICMaPI + DII + DIETI 1224 5000 + Vincenzo Russo fluidized bed + Maria Chiara Mosca chemical reactors; Bologna DICAM + DEI + DIN 4000 + DICAM Termofluidodinamica + Pier Luigi Dragoni HVAC + Massimo Cocconcelli geotechnics sedimentation; Padova DII + DEI + ICEA 3000 + CISAS + Massimo Brunetti drying agro-food + Marco Sartori sedimentation hydraulics; Pisa DICI + DESTeC 2500 + Walter Grassi multiphase + Andrea Lucchini drying; Politecnico Bari DEI 2500; Sapienza Roma DICEA + DIMA + DIET 4000 + Marco Cecconi HVAC + Antonio Liaci spray; Genova DICCA + DIBRIS 2000 + IIT 1700; Trento DICAM 1962 + Lorenzo Brentari sedimentation + Marco Tubino river; Salerno DIIN 1968; Firenze DIEF 1321; Cagliari DIMCM 1606; Universita Pisa DICI + Galileo legacy.

Italia industria sedimentazione + acque depurazione: Acea Roma 1909 12000 + acquedotto Capitolino + depurazione + sedimentazione primaria/secondaria; Iren Torino 1907 7800 + Genova + Reggio Emilia + Parma + acque + depurazione; A2A Milano + Brescia 12000 + acque urbane sedimentazione; Veritas Venezia 2000 + acque Lagoon Venezia + sedimentazione; Hera SpA Bologna 9000; SMAT Torino 1100.

Italia industria fluidized bed + biomass + cogenerazione: Eni SpA Roma 1953 32000 + raffinerie 5 sites + FCC fluidized catalytic cracking Sannazzaro Livorno Taranto Gela Sarroch + Versalis polimeri; Saipem Milano 1957 32000 oil&gas EPC + offshore + FCC reactor; Tecnimont Milano Maire Tecnimont 6000 EPC + biomass gasification fluidized bed; Snam Milano 6000; Falck Renewables Sesto San Giovanni 1500 + biomass; Itea Italgreen biomass.

Italia industria powder technology + farmaceutica: Chiesi Farmaceutici Parma 1935 7000 + powder DPI dry powder inhalers; Recordati Milano 1926 4500; Zambon 2500; Angelini Pharma Roma 5000; Menarini Firenze 17000; Bracco Diagnostica Milano 3500 + powder contrast media; Italfarmaco 3000; Dompe 800.

Italia industria cementi + powder + minerals: Italcementi Bergamo 1864 + HeidelbergCement 2016 22000 + cement powder + Carbon8 CCS + Italgen + Calcestruzzi; Buzzi Unicem Casale Monferrato 1907 9500; Cementi Rossi Caorso 1500; Colacem Gubbio 1962 800; Mapei SpA Milano 1937 10000 + adhesives + admixtures cement; Pizzarotti Group Parma 1910 1500; Webuild SpA (Salini Impregilo) Milano 14000; Buzzi Unicem cement plants Augusta Vibo Valentia Settimello + Italian Riviera; Cementir Rome 1947 3300; Sika Schweiz Italia 1500.

Italia industria mining + minerals + sedimentation: Eni SpA Mineraria 32000; Italian Quarries Marble Carrara 200 + Statuario Bardiglio Cipollino marble; Salini Impregilo Webuild 14000 + tunnels mining; SDIRO Sardegna lead-zinc; Iglesiente Sulcis Sardegna mining historical.

Italia industria automotive + ball bearings + powder metallurgy: Brembo Stezzano BG 1961 14000 + brake disc cast iron + powder metallurgy + ball bearings; Magneti Marelli (Marelli) 39000; Ferrari Maranello 5500 + ball bearings; SKF Italia + Bergamo + Carraro Padova 1933 1500 ball bearings; Carraro Padova 1932 4000 + transmissions + bearings; OZ Group San Martino Buon Albergo VR 1971 1500; FAVA SpA Roma 250; Riva Forni Elettrici 5000.

Italia industria spazio + Galileo navigation: Italia ESA Galileo navigation system Brussels HQ + Italy contribution + Telespazio Roma + Thales Alenia Space Roma + Galileo IOV satellites + Galileo FOC + 30 satellites + Italy ranking ESA contribution; Telespazio Roma 1961 + Leonardo+Thales JV 2700 + Galileo ground control + Fucino space center; Thales Alenia Space Italia 7800 + Galileo IOV/FOC satellites construction + Cosmo-SkyMed + Sentinel; e-Geos 600 Cosmo-SkyMed; ASI Roma 1988 250 + Galileo + ESA contribution.

Formula numero Galileo

Definizione: Ga = g * L^3 / nu^2. Equivalente Ga = Re^2 a velocità terminale Stokes regime.

Archimedes: Ar = ((rho_p - rho_f)/rho_f) * Ga. Correzione densità particella/fluido.

Stokes terminal velocity: v_t = (rho_p-rho_f)*g*L^2/(18*nu*rho_f) + Stokes 1851 + Re_t<0.1.

Drag coefficient regimi: Stokes Re<0.1 CD=24/Re; intermediate 0.1<Re<1000 Schiller-Naumann CD=(24/Re)*(1+0.15*Re^0.687); Newton 1000<Re2.5e5 CD~0.18-0.2.

Geldart classification: A aeratable 30-100 micron + smooth fluidization (FCC); B sand-like 100-1000 + bubbles; C cohesive 1000 micron + grains coal.

Storia e contesto

Galileo Galilei (Pisa Italia 1564 - Arcetri Florence 1642) Universita Pisa medicine 1581 + matematica + Padova Universita 1592-1610 chair geometry mechanics astronomy + Firenze Granducato Toscana 1610-1633 + casa di Arcetri 1634-1642 sotto arresto domiciliare; 1589-1592 esperimenti caduta libera Torre Pendente di Pisa (leggendari, contestati storici ma supportati Stillman Drake biographer) + Galileo dimostra velocità di caduta non dipende dalla massa rifiuta Aristotele + 'De motu' 1592 unpublished manuscript + 'Discorsi e Dimostrazioni Matematiche intorno a Due Nuove Scienze' 1638 Leiden Elsevier 'Two New Sciences' classic + caduta libera + moto uniformemente accelerato + parabola proiettile + Galileo pioneer scientific method + telescope 1609 + Jupiter moons Io Europa Ganymede Callisto Sidereus Nuncius 1610 + Sun spots + Saturn rings (anelli) + Venus phases + Inquisition trial 1633 abjuration heliocentric Copernicus 'Dialogo sopra i due massimi sistemi del mondo' 1632 + house arrest Arcetri 1633-1642 + 'eppur si muove' apocryphal + Tycho Brahe Kepler corrispondenza + Salviati friend Discorsi character;

Padova Universita 18 anni 1592-1610 chair geometria meccanica astronomia + Galileo Galilei Universal Clock + telescopi + telescope refractor + 1610 'Sidereus Nuncius' Venice telescope discoveries + 'Dialogo' 1632 Florence Landini classic dialogue Aristotelean Ptolemaic vs Copernican Salviati Sagredo Simplicio characters; lascia Discorsi 1638 Leiden secret manuscript published Elsevier outside Italy.

George Gabriel Stokes (Skreen Sligo Ireland 1819 - Cambridge UK 1903) Cambridge University Pembroke College + Lucasian Professor of Mathematics 1849-1903 (succeeding Joshua King + before P A M Dirac 1932); 1851 paper Trans Cambridge Phil Soc 9:8 'On the effect of the internal friction of fluids on the motion of pendulums' classic Stokes' law drag formula F_d = 6*pi*mu*v*r sphere viscous fluid + Stokes drag + 1845 paper Trans Cambridge Phil Soc 8:287 'On the theories of the internal friction of fluids in motion' Navier-Stokes equations + 1850 paper 'On the change of refrangibility of light' fluorescence Stokes shift + 1854 paper 'On the dynamical theory of diffraction' wave optics; Royal Society President 1885-1890 + Smith Prize 1841 + Copley Medal 1893 + Lucasian Professor + classical mathematical physics + Cambridge classic.

L Schiller + A Z Naumann Germany 1933; 1933 paper Z Verein Deut Ing 77:318 'Uber die grundlegenden Berechnungen bei der Schwerkraftaufbereitung' Schiller-Naumann correlation drag coefficient intermediate Reynolds number 0.1<Re<1000 + classic.

Sabri Ergun (Turkey 1918 - 2006) PennState University Chemistry + Argonne National Laboratory + Drexel University; 1952 paper Chem Eng Prog 48:89 'Fluid flow through packed columns' Ergun equation packed bed + viscous Kozeny-Carman + inertia Burke-Plummer + standard reference fluidized bed minimum fluidization.

Derek Geldart (UK 1929-) Bradford University + Cambridge University; 1973 paper Powder Technol 7:285 'Types of gas fluidization' Geldart classification A/B/C/D + powder technology + fluidized bed regime + 30 micron 100 micron 1000 micron boundaries.

Chien-Yu Wen + Y H Yu West Virginia University; 1966 AIChE J 12:610 'Mechanics of fluidization' Wen-Yu correlation minimum fluidization velocity U_mf + Ergun + standard reference.

D Kunii + Octave Levenspiel 'Fluidization Engineering' 1991 textbook classic. Eredita: Crowe-Sommerfeld-Tsuji 'Multiphase Flows with Droplets and Particles' 2nd ed 2011 CRC Press 509 pagine + Kunii-Levenspiel 'Fluidization Engineering' 2nd ed 1991 491 pagine + Yang 'Handbook of Fluidization' 2003 CRC + Davidson-Harrison 'Fluidization' 2nd ed 1971 Academic Press classic; Clift-Grace-Weber 'Bubbles Drops Particles' 1978 Academic Press classic 380 pagine.

Italia ricerca sedimentazione + fluidized bed + powder + multiphase: CNR-IIA Istituto Inquinamento Atmosferico Monterotondo Roma + Sesto Fiorentino 250 + particolato atmospheric PM10 PM2.5 sedimentation Stokes; CNR-ICMATE Istituto Chimica Materia Condensata Tecnologie Energetiche Genova + Padova + Lecco 200 + powder technology + crystal growth; CNR-ITAE Istituto Tecnologie Avanzate Energia Messina 200 + fluidized bed combustion biomass + heat pumps; ENEA Agenzia Nazionale Nuove Tecnologie Energia Roma + Casaccia + Frascati + Brindisi + Trisaia + 11 centri 2700 + biomass fluidized bed gasification BFB CFB + thermal storage; RSE Ricerca Sistema Energetico Milano 1000 + fluidized bed combustion CFB Circulating Fluidized Bed; SOGIN SpA Roma 1100 nuclear decommissioning + radioactive waste fluidized + air sedimentation; OGS Trieste 250 + ocean sedimentation Adriatico Mediterraneo; CMCC Lecce 200; INFN Italia Frascati LNF 800 + Roma + L'Aquila + Catania + Pisa + Trieste + Genova 5000; CNR-ITC Tecnologia Costruzioni Bari + Padova 200; CNR-ISMAR Venezia + Trieste + Bologna + Pozzuolo 200 ocean sedimentation.

Italia università Galileo + sedimentazione + fluidized bed: Politecnico Milano DEIB + DENG + LCM 7500 + Carlo Cercignani Boltzmann legacy 1939-2010 kinetic theory + Renzo Piazza CFD multiphase + Pier Luigi Bonomi multiphase fluidized bed + Marco Mauri thermal storage; Politecnico Torino DENERG + DIATI 1859 5000 + Marco Fiori multiphase + Antonio Tomassetti fluidized bed CFB biomass; Federico II Napoli DICMaPI + DII + DIETI 1224 5000 + Vincenzo Russo fluidized bed + Maria Chiara Mosca chemical reactors + Pasquale Brescia HVAC; Bologna DICAM + DEI + DIN 1088 4000 + DICAM Termofluidodinamica + Pier Luigi Dragoni HVAC + Massimo Cocconcelli geotechnics sedimentation soil mechanics; Padova DII + DEI + ICEA 1222 3000 + CISAS + Renato Lazzarin HVAC + Massimo Brunetti drying agro-food + Marco Sartori sedimentation hydraulics fluvial estuary; Pisa DICI + DESTeC 1343 2500 + Walter Grassi multiphase + Andrea Lucchini drying + Daniele Testi heat pumps; Politecnico Bari DEI 1990 2500; Sapienza Roma DICEA + DIMA + DIET 1303 4000 + Marco Cecconi HVAC + Antonio Liaci spray combustion; Genova DICCA + DIBRIS 1481 2000 + IIT 1700; Trento DICAM 1962 + Lorenzo Brentari sedimentation + Marco Tubino river hydraulics + Aronne Armanini sediment transport; Salerno DIIN 1968;

Firenze DIEF 1321; Cagliari DIMCM 1606; Universita Pisa DICI + Galileo Galilei legacy + birthplace Pisa 1564 + Universita di Pisa medicina 1581.

Italia industria sedimentazione + acque depurazione + acquedotti: Acea Roma 1909 12000 + acquedotto Marcio + Capitolino + Acqua Lazio + depurazione + sedimentazione primaria/secondaria; Iren Torino 1907 7800 + Genova + Reggio Emilia + Parma + acque + depurazione; A2A Milano + Brescia 12000 + acque urbane sedimentazione; Veritas Venezia 2000 + acque Lagoon Venezia + Acqua Alta sedimentazione; Hera SpA Bologna 9000 + multiservizio Emilia-Romagna; SMAT Torino 1100 + Società Metropolitana Acque Torino; Acquedotto Pugliese Bari 1919 1900 + Apulia water supply; Acque del Veneto 1500; Acque Bresciane 700; Sorgenia 200.

Italia industria fluidized bed + biomass + cogenerazione + FCC: Eni SpA Roma 1953 32000 + raffinerie 5 sites Sannazzaro de'Burgondi PV + Livorno + Taranto + Gela CL + Sarroch CA + FCC fluidized catalytic cracking + Versalis polimeri; Saipem Milano 1957 32000 oil&gas EPC + offshore + FCC reactor design; Tecnimont Milano Maire Tecnimont 6000 EPC + biomass gasification fluidized bed; Snam Milano 6000; Falck Renewables Sesto San Giovanni 1500 + biomass; Italgreen biomass; Polimeri Europa Versalis Eni 5000; API Group Roma 1933 1100 + raffinerie Falconara Marittima; IPLOM SpA Busalla GE 1947 200.

Italia industria powder technology + farmaceutica + DPI inhalers: Chiesi Farmaceutici Parma 1935 7000 + DPI dry powder inhalers + Foster + Trimbow + Atimos + Aerivio; Recordati Milano 1926 4500; Zambon Bresso MI 2500; Angelini Pharma Roma 5000; Menarini Firenze 17000 leading Italian pharma; Bracco Diagnostica Milano 3500 + powder contrast media imaging; Italfarmaco Milano 3000; Dompe Farmaceutici 800.

Italia industria cementi + calcestruzzo + admixtures: Italcementi Bergamo 1864 + HeidelbergCement 2016 22000 + cement powder + Carbon8 CCS + Italgen + Calcestruzzi; Buzzi Unicem Casale Monferrato AL 1907 9500 + cement plants Italy 12 + 5 Germany + Czech + Russia + Mexico + USA; Cementi Rossi Caorso PC 1500; Colacem Gubbio PG 1962 800; Mapei SpA Milano 1937 10000 + adhesives + admixtures cement + grout + dry mortar; Sika Schweiz Italia 1500 + admixtures; Pizzarotti Group Parma 1910 1500; Webuild SpA (Salini Impregilo) Milano 14000 + tunnels Stretto di Messina; Cementir Rome 1947 3300.

Italia industria mining + marble Carrara + minerals: Italian Marble Carrara 200 quarries 12000 + Statuario + Bardiglio + Cipollino marmo bianco; Salini Impregilo Webuild 14000 + tunnels mining + Mont-Blanc; SDIRO Sardegna 1500 lead-zinc + barium + fluorite mining; Iglesiente Sulcis Sardegna mining historical 5000+ years Phoenician Roman lead silver.

Italia industria spazio + ESA Galileo navigation: ESA European Space Agency Galileo + Italy contribution + 30 satellites + IOV In-Orbit Validation + FOC Full Operational Capability + Ariane Vega launch + Galileo Reference Time + 1 ns timing accuracy + worldwide coverage 2024 + EGNOS European Geostationary Navigation Overlay Service; Telespazio Roma 1961 + Leonardo+Thales JV 2700 + Galileo ground control + Fucino space center; Thales Alenia Space Italia Roma + Torino + Milano 1962 7800 + Galileo IOV/FOC satellites construction + Cosmo-SkyMed + Sentinel + ISS modules; e-Geos Roma 600 + Cosmo-SkyMed Earth observation; ASI Roma 1988 250 + Galileo + ESA Italian contribution; Avio Colleferro Roma 5000 + Vega Vega-C launcher + Ariane 6 P120C solid rocket Galileo launches.

Tabella valori Galileo

• Ga = g*L^3/nu^2: definizione
• Ga = Re^2 terminal Stokes: equivalente
• Ar = (rho_p-rho_f)/rho_f * Ga: Archimedes
• Re_t = sqrt(Ga/18) Stokes: terminale
• Stokes v_t = (rho_p-rho_f)*g*L^2/(18*nu*rho_f)
• Ga<1 Stokes regime: clay colloidi
• 1<Ga<1000 intermediate: sand fine
• 1000<Ga<1e5 Newton: sand spheres
• Ga>1e5 high inertia: boulders
• Stokes drag CD=24/Re: Re<0.1
• Schiller-Naumann CD=(24/Re)*(1+0.15*Re^0.687): 0.1-1000
• Newton CD=0.44: 1000-2.5e5
• Crisi resistenza CD=0.18: Re>2.5e5
• Geldart A aeratable: 30-100 micron FCC
• Geldart B sand-like: 100-1000 bubbles
• Geldart C cohesive: <30 cement flour
• Geldart D spoutable: >1000 grains
• Galileo Italia 1564-1642: Pisa Tower 1589
• Discorsi 1638 Leiden: Two New Sciences
• Stokes 1851: drag formula F=6*pi*mu*v*r
• Schiller-Naumann 1933: drag intermediate
• Ergun 1952: packed bed
• Geldart 1973: classification
• Wen-Yu 1966: U_mf fluidization
• Italia ENI 32000: FCC raffinerie
• Italia Saipem 32000: FCC reactor
• Italia ENEA 2700: biomass fluidized bed
• Italia Italcementi 22000: cement powder
• Italia Mapei 10000: admixtures
• Italia Acea 12000: depurazione
• Italia Galileo navigation: ESA Italy contribution

Esempi numerici dettagliati

Esempio 1 - Argilla particella in acqua. g=9.81, L=1e-6 m (1 micron clay), nu=1e-6 m^2/s acqua. Ga = 9.81*1e-18/1e-12 = 9.81e-6. Stokes regime + v_t = 9.8e-13 m/s extremely slow + sedimentazione clay laghi. Italia: Acea Roma 12000 + sedimentazione primaria depurazione.

Esempio 2 - Sabbia in acqua. g=9.81, L=1e-3 m (1 mm sand), nu=1e-6. Ga = 9.81*1e-9/1e-12 = 9810. Newton regime + v_t ~ 0.1 m/s. Italia: Iren Torino + Acea + sedimentazione fiumi Po Tevere Arno.

Esempio 3 - Pellet biomass in aria. g=9.81, L=3e-3 m, nu=1.5e-5 aria. Ga = 9.81*2.7e-8/2.25e-10 = 1177. Newton fluidized bed biomass. Italia: ENEA Casaccia 2700 + RSE 1000 + biomass CFB Circulating Fluidized Bed.

Esempio 4 - Ciottolo in acqua. g=9.81, L=5e-2 m (5 cm pebble), nu=1e-6. Ga = 9.81*1.25e-4/1e-12 = 1.23e8. High inertia + Re_t ~ 100000. Italia: Salini Impregilo 14000 + tunnels + sedimentation hydraulics.

Esempio 5 - Goccia olio in acqua. g=9.81, L=1e-4 m, nu=1e-6. Ga = 9.81*1e-12/1e-12 = 9.81. Intermediate + emulsion separation. Italia: Eni 32000 + raffinerie + oil-water separation Sannazzaro Livorno.

Limiti e considerazioni

1. Sphere assumption: real particles non-sferiche + shape factor + Heywood factor.
2. Concentration: hindered settling Richardson-Zaki + concentrazione modify v_t.
3. Wall effects: Faxen correction + velocity profile + Re piccolo.
4. Inertia non-Newtonian: shear-thinning + viscoelastic + drag CD modify.
5. Compressibility: high speed + Mach + drag varies.
6. Turbulence ambient: Stokes vs ambient turbulent + scale separation.
7. Brownian motion: <1 micron + thermal Brownian + Peclet small.
8. Surface tension: bubble droplet vs solid + Marangoni + Bond.
9. Multiple particles: Stokes law fail + concentrazione + cluster sedimentation.
10. Magnetic field: ferrofluid + magnetic separation + drag MHD.

Errori comuni

Confondere Ga e Ar: Ga gravity/viscous; Ar = Ga * (rho_p-rho_f)/rho_f density correction.

Stokes regime limits: Re_t<0.1 strict + 0.11000 Newton.

Sphere assumption: real non-sferiche + Heywood shape factor + ellipsoid + cylinder modifies.

Hindered settling: Richardson-Zaki concentration > 1% + v_t reduce.

Wall effects: Faxen correction + container small + Re piccolo Stokes + parete.

Applicazioni pratiche

1. Acque depurazione sedimentazione primaria/secondaria: Italia Acea Roma 12000 + Iren Torino 7800 + A2A 12000 + Veritas Venezia 2000 + Hera Bologna 9000 + SMAT Torino 1100 + Acquedotto Pugliese 1900.
2. Fluidized bed FCC catalytic cracking raffinerie: Italia Eni 32000 raffinerie 5 sites Sannazzaro Livorno Taranto Gela Sarroch + Saipem 32000 reactor + Tecnimont 6000 EPC.
3. Biomass gasification fluidized bed CFB: Italia ENEA Casaccia 2700 + RSE Milano 1000 + Falck Renewables 1500 + Italgreen + Tecnimont biomass.
4. Powder technology pharma DPI inhalers: Italia Chiesi Parma 7000 DPI Foster Trimbow + Recordati 4500 + Zambon + Angelini 5000 + Menarini 17000 + Bracco 3500 + Italfarmaco + Dompe.
5. Cementi powder admixtures: Italia Italcementi HeidelbergCement 22000 + Buzzi Unicem 9500 + Mapei 10000 admixtures + Cementi Rossi + Colacem + Cementir 3300.
6. Mining marble Carrara: Italia Carrara 200 quarries 12000 marble Statuario + Salini Impregilo 14000 + SDIRO Sardegna lead-zinc + Iglesiente Sulcis.
7. Sediment transport rivers: Italia ARPA 21 regional 6000 + Universita Trento Brentari Tubino Armanini + Po Tevere Arno + sediment transport hydraulics.
8. Powder metallurgy automotive ball bearings: Italia Brembo Stezzano 14000 brake powder + SKF Italia + Carraro Padova 4000 + OZ Group + Marelli 39000.
9. Food powder spray drying coffee milk: Italia Lavazza 4000 + Granarolo 2500 + Illy 1300 + Ferrero 38000 + Galbani 1500 + Parmigiano Consortium 25000.
10. Galileo navigation system ESA Italy: ESA European Space Agency 30 satellites + Telespazio Roma 2700 ground control Fucino + Thales Alenia Space 7800 satellites construction + e-Geos 600 + ASI 250 + Avio 5000 Vega launches Galileo.

Aggiornato a luglio 2026