Come calcolare il numero di Eotvos (Eo) noto anche come Bond number (Bo) rapporto adimensionale fluidodinamica multiphase che misura il rapporto tra forze gravitazionali (peso differenziale tra due fasi a densità diverse) e forze di tensione superficiale (capillarità interfacciale): Eo = (rho_h - rho_l) * g * L^2 / sigma. rho_h densità fase pesante kg/m^3 (acqua 1000 a 20°C); rho_l densità fase leggera kg/m^3 (aria 1.2 a 20°C; vapore 0.6); g accelerazione gravita 9.81 m/s^2 standard Earth (1.62 m/s^2 Luna; 3.71 m/s^2 Marte); L lunghezza caratteristica m (diametro bolla, droplet, capillary tube); sigma tensione superficiale interfacciale N/m (acqua-aria 0.0728 N/m a 20°C; mercurio-aria 0.486 N/m; olio-acqua 0.025-0.05 N/m).

Lunghezza capillare L_c = sqrt(sigma / (delta_rho * g)) lunghezza caratteristica naturale; acqua-aria 20°C L_c = 2.71 mm; mercurio-aria 20°C L_c = 1.91 mm; olio-acqua L_c = 1-2 mm. Eo = (L/L_c)^2 quadrato rapporto.

Classificazione regime: Eo<0.1 surface tension dominated sferica droplet/bubble piccoli; 0.1<Eo<1 capillary scale small distortion; 1<Eo<10 mixed regime ellipsoidal bubble; 10<Eo<100 gravity dominated cap/spherical-cap bubble; Eo>100 strong gravity large bubble flat interface tipico boilers + reactors industrial. Storia: Lorand Eotvos (Hungary 1848-1919) Universita Budapest fisico ungherese pioneer geofisica + capillarità 1886 paper Wiedemann's Annalen on surface tension ratio + Eotvos rule capillarity surface tension + temperature relationship sigma = K_E * (T_c - T) * (M/rho)^(2/3).

Wilfrid Norman Bond (UK 1897-1937) Cambridge University fisico inglese 1928 paper Phil Mag 'Bubble in a viscous liquid' systematic dimensional analysis multiphase flow + Bond number eponym for surface tension ratio.

Italia ricerca + università: Politecnico Milano DEIB Dipartimento Elettronica Informazione Bioingegneria 1863 7500 students + DENG Dipartimento Energia + DCMC Chimica Materiali Chimica Industriale + LCM Laboratorio Microreattori; Politecnico Torino DENERG Energia 1859 5000 + DIATI Ambiente Territorio Infrastrutture; Sapienza Roma DICMA Ingegneria Chimica Materiali Ambiente 1303 4000 + DIMA Meccanica Aerospaziale; Federico II Napoli DICMaPI Ingegneria Chimica 1224 5000 + DII; Bologna DICAM Civile Chimica 1088 4000 + DEI; Padova DII 1222 3000 + ICEA; Pisa DICI 1343 2500 + DESTeC; Politecnico Bari DIChim 1990 2500 + DEI; Trento DICAM 1962; Salerno DIIN 1968; Genova DICCA 1481; Firenze DIEF 1321. Italia istituzioni ricerca: CNR-INM Istituto Marina Roma + Genova 200 idrodinamica navale + multiphase flow + bubble dynamics; CNR-IRC Istituto Ricerche Combustione Napoli 1980 200 combustione + droplet dynamics + spray atomization + atomization injectors; CNR-IPCB Polimeri Compositi Biomateriali Napoli + Pozzuoli 250; ENEA Roma 1960 2500 + Frascati ITER + Brasimone Bologna nuclear; INFN Roma 1951 5000 + LNF Frascati.

Italia industria petrolchimica + chimica + bubble columns: Eni SpA Roma 1953 32000 + Eni R&D San Donato Milanese 1500 + raffinerie Sannazzaro Pavia + Taranto + Livorno + Venezia + Gela Sicilia 5 raffinerie + Versalis polimeri Mantova Ferrara Brindisi Priolo Gargallo + Plenitude rinnovabili; Saipem Milano 1957 32000 oil&gas EPC engineering procurement construction + offshore platforms + subsea pipelines + drilling rigs + bubble flow analysis multiphase; Tecnimont Milano Maire Tecnimont 6000 EPC chemical petrochemical + bubble columns reactor design; Ansaldo Energia Genova 1853 3000 turbine gas + steam + boilers + nuclear; De Nora Industrie Milano 1923 2000 chlor-alkali electrochemistry + diaphragm membrane + green hydrogen electrolyzers + bubble dynamics gas evolution; BASF Italia Cesano Maderno Monza + Solvay Italia Bollate + Versalis Eni petrochemical + reactor; Edison Milano 1884 utility 5000; Snam Milano 1941 6000 gas grid 30000 km gasdotti pipelines two-phase flow oil-water-gas slug + churn + annular regime; ENI Plenitude 4500 + Iren 9000 + A2A 13000 utility. Italia industria oil&gas pipelines + multiphase flow: Saipem Castorone Castor 6 + offshore Mediterraneo platforms + Adriatic + ENI Plenitude + Eni Mediterraneo gas pipelines + Snam 30000 km gasdotti two-phase flow analysis stratified slug churn annular regime;

ANIE Federazione Imprese Elettrotecniche Roma. Italia industria farmaceutica drug delivery + spray + droplet: Bracco Imaging Milano 3700 microbubble ultrasound contrast agents + microfluidica; Chiesi Farmaceutici Parma 5500 respiratory drug aerosol delivery inhalers MDI metered dose + DPI dry powder; Recordati Milano 4500 cardiovascular; Boehringer Ingelheim Italia + GSK Italia inhalers + nasal sprays. Italia ricerca microfluidica + droplet: Politecnico Milano DEIB LCM Laboratorio Microreattori + STMicroelectronics Catania microfluidica MEMS + Sant'Anna Pisa biomechanics microfluidics + IIT Italian Institute Technology Genova 1700 + IFC Istituto Fisiologia Clinica CNR Pisa.

Formula numero Eotvos (Bond)

Definizione: Eo = Bo = (rho_h - rho_l) * g * L^2 / sigma rapporto adimensionale gravita/tensione superficiale. delta_rho = rho_h - rho_l differenza densità kg/m^3 (heavy - light phase); g 9.81 m/s^2 Earth; L lunghezza caratteristica m; sigma tensione superficiale interfacciale N/m.

Lunghezza capillare: L_c = sqrt(sigma / (delta_rho * g)) lunghezza caratteristica naturale tensione superficiale. Acqua-aria 20°C L_c=2.71 mm; mercurio-aria 1.91 mm; olio-acqua 1-2 mm. Eo = (L/L_c)^2 quadrato rapporto.

Surface tension typical: acqua-aria 20°C sigma=0.0728 N/m; acqua-aria 80°C sigma=0.0626; etanolo-aria 20°C 0.0223; mercurio-aria 20°C 0.486; olio motore-aria 0.030; vapore-acqua P=1 atm 0.0589 a 100°C; CO2 supercritical-acqua 0.001-0.005.

Significato fisico: Eo<>1 gravita domina + bolle ascensione + Stokes regime + drag balance buoyancy.

Numeri correlati: Reynolds Re=u*L/nu; Weber We=rho*u^2*L/sigma inerzia/tensione; Capillary Ca=mu*u/sigma viscosità/tensione; Morton Mo=g*mu^4*delta_rho/(rho_h^2*sigma^3) bubble shape regime; Galileo Ga=rho^2*g*L^3/mu^2.

Storia e contesto

Lorand Eotvos (Pest Ungheria 1848 - Budapest 1919) Universita Heidelberg + Konigsberg + Universita Budapest professor 1872-1919, fisico ungherese pioneer geofisica + Roland Eotvos Geophysical Institute Budapest fondazione + 1886 paper Wiedemann's Annalen 'Uber den Zusammenhang der Oberflachenspannung der Flussigkeiten mit ihrem Molekularvolumen' su capillarità + tensione superficiale + Eotvos rule sigma * V^(2/3) = K_E * (T_c - T) costante Eotvos K_E = 2.1e-7 J/(K*mol^(2/3)) per liquidi non-polari; 1890 Eotvos pendulum torsion balance esperimenti gravity Princess Maud 1909 + Hungary geological survey + RIN ricerca petrolio Romania 1910s.

Wilfrid Norman Bond (UK 1897-1937 Cambridge) fisico inglese 1928 paper Philosophical Magazine 'Bubble in a viscous liquid' systematic dimensional analysis multiphase flow ratio surface tension to gravity buoyancy; Bond number eponym; muore young Cambridge 40 anni.

Eredita: Clift-Grace-Weber 1978 'Bubbles, Drops, and Particles' Academic Press classic textbook bubble dynamics + Levich 1962 'Physicochemical Hydrodynamics' Prentice Hall + de Gennes-Brochard-Quere 2002 'Capillarity and Wetting Phenomena' Springer + Pierre-Gilles de Gennes Nobel Physics 1991.

Italia ricerca + università ingegneria chimica: Politecnico Milano DEIB + DENG Energia + DCMC Chimica Materiali Chimica Industriale + LCM Laboratorio Microreattori 7500 students professori Renato Rota chemical engineering + Massimo Morbidelli reactor design ETH Zurich + Davide Manca process safety + scuola Carlo Cercignani Boltzmann legacy; Politecnico Torino DENERG + DIATI 1859 5000 students; Sapienza Roma DICMA + DIMA 1303 4000 students bioingegneria; Federico II Napoli DICMaPI + DII 1224 5000 students; Bologna DICAM + DEI 1088 4000 students; Padova DII + ICEA 1222 3000 students; Pisa DICI + DESTeC 1343 2500 students; Politecnico Bari DIChim + DEI 1990 2500 students; Trento DICAM 1962 + DISI; Salerno DIIN 1968 Fisciano; Genova DICCA + DIBRIS 1481 + IIT 1700; Firenze DIEF 1321; Cagliari DIMCM 1606; Universita Brescia + Trieste + Napoli II Caserta + Catania + Palermo.

Italia CNR + ENEA + INFN: CNR-INM Istituto Marina Roma + Genova 200 idrodinamica navale + multiphase flow + bubble dynamics + INSEAN Roma test ship hull resistance bacino 470 m; CNR-IRC Istituto Ricerche Combustione Napoli 1980 fondato 200 employees combustione kinetics chimica + droplet dynamics atomization spray injectors + flame propagation + emission control NOx CO HC + biomass biofuels biodiesel + Politecnico Milano DENG partner; CNR-ICCOM Istituto Chimica Composti Organometallici Firenze 250 catalisi; CNR-ISOF Istituto Sintesi Organica Fotoreattivita Bologna 200 photocatalysis green chemistry; CNR-IPCB Polimeri Compositi Biomateriali Napoli + Pozzuoli + Cagliari 250 polymers + biomedical; CNR-IIA Inquinamento Atmosferico Monterotondo Roma 250 + atmospheric chemistry; ENEA Agenzia Nazionale Nuove Tecnologie Energia Sviluppo Economico Sostenibile Roma EUR 1960 2500 employees energia + nucleare + rinnovabili + smart grid + biomasse + idrogeno + ITER ENEA Frascati + BRASIMONE Brasimone Bologna nucleari + Casaccia Roma + Saluggia Vercelli + ENEAGRID Cresco-Cresco6 1.7 PetaFLOPS HPC; INFN Roma 1951 5000 + Frascati LNF 800 DAFNE collider beam dynamics + ultra-high vacuum UHV; INRIM Torino 2006 250 metrologia.

Italia industria petrolchimica + chimica + reactors bubble columns: Eni SpA Roma 1953 32000 + Eni R&D San Donato Milanese 1500 + raffinerie Sannazzaro Pavia + Taranto + Livorno + Venezia + Gela Sicilia 5 raffinerie + Versalis polimeri Mantova Ferrara Brindisi Priolo Gargallo + Plenitude rinnovabili 7 milioni clienti + Renewable Energy biotech; Saipem Milano 1957 32000 oil&gas EPC engineering procurement construction + offshore platforms Castor Castorone CMS Saipem 7000 + subsea pipelines + drilling rigs + dynamic positioning DP3 + multiphase flow analysis bubble flow stratified slug churn annular regime; Tecnimont Milano Maire Tecnimont 6000 EPC chemical petrochemical reactor design bubble columns + heat exchanger; Ansaldo Energia Genova 1853 3000 turbine gas + steam + boilers + nuclear + heat exchanger; De Nora Industrie Milano 1923 2000 chlor-alkali electrochemistry + diaphragm membrane technology + green hydrogen electrolyzers PEM AEM AECN + bubble dynamics gas evolution H2 O2 hydrogen oxygen production hydrolysis; BASF Italia Cesano Maderno Monza chemical + heat exchanger + reactor; Solvay Italia Bollate Milano + Rosignano Solvay Livorno chemical + heat exchanger + reactor; Versalis Eni Mantova + Ferrara + Brindisi + Priolo Gargallo Sicilia petrochemical + heat exchanger + bubble columns styrene rubber elastomeri; Sasol Italia Augusta Siracusa petrochemical; Lanxess Filago Bergamo + Bayer Italia Milano chemical + reactor.

Italia industria oil&gas pipelines + multiphase: Snam Milano 1941 6000 gas grid 30000 km gasdotti pipelines two-phase flow oil-water-gas + offshore Mediterraneo + Adriatic + Eni Plenitude utility 4500 + Iren Reggio Emilia 9000 + A2A Milano Brescia 13000. Italia industria farmaceutica drug delivery spray atomization: Bracco Imaging Milano 3700 microbubble ultrasound contrast agents + interventional radiology + medical imaging; Chiesi Farmaceutici Parma 5500 respiratory drug delivery aerosol pMDI pressurized metered dose inhalers + DPI dry powder inhalers Trimbow Foster Atimos; Recordati Milano 4500 cardiovascular; Menarini Florence pharmaceutical + diagnostics; Boehringer Ingelheim Italia Milano respiratory pMDI Spiriva tiotropium + Striverdi olodaterol; GSK Italia Verona respiratory pMDI Ventolin + Seretide salmeterol fluticasone DPI; AstraZeneca Italia Milano respiratory Symbicort budesonide formoterol DPI Turbuhaler; Italfarmaco Milano + Alfasigma Bologna 3000 pharmaceutical; Zambon Bresso Milano 2500 respiratory; Dompe Milano + L'Aquila pharmaceutical immunology. Italia ricerca microfluidica + droplet biomedicale: Politecnico Milano DEIB LCM Laboratorio Microreattori microfluidica + droplet generation T-junction microfluidic chip; STMicroelectronics Catania microfluidica MEMS Lab-on-Chip; Sant'Anna Pisa biomechanics microfluidics 1987 fondata; IIT Italian Institute Technology Genova 1700 + IFC Istituto Fisiologia Clinica CNR Pisa cardiovascular;

Sapienza Roma DICMA + DIMA + DIET microfluidica.

Tabella valori Eotvos Bond

• Eo = Bo = delta_rho * g * L^2 / sigma
• L_c = sqrt(sigma/(delta_rho*g)): capillary length
• Acqua-aria 20°C L_c: 2.71 mm
• Mercurio-aria L_c: 1.91 mm
• Olio-acqua L_c: 1-2 mm
• Acqua-aria 20°C sigma: 0.0728 N/m
• Mercurio-aria sigma: 0.486 N/m
• Etanolo-aria sigma: 0.0223 N/m
• Olio motore-aria: 0.030 N/m
• Eo<0.1 surface tension: sferica drop/bubble
• 0.1<Eo<1 capillary: small distortion
• 1<Eo<10 mixed: ellipsoidal bubble
• 10<Eo<100 gravity: cap/spherical-cap
• Eo>100 strong gravity: large flat interface
• Bubble 1 mm acqua-aria: Eo=0.13
• Bubble 5 mm acqua-aria: Eo=3.4
• Bubble 50 mm acqua-aria: Eo=336
• Eotvos 1886 Wiedemann Annalen: Hungary
• Bond 1928 Phil Mag: UK Cambridge
• Clift-Grace-Weber 1978: Academic Press
• de Gennes Nobel 1991: Capillarity
• Italia Politecnico Milano DEIB: 7500 + LCM
• Italia CNR-INM Roma INSEAN: 200 470 m bacino
• Italia CNR-IRC Napoli 1980: 200 combustion droplet
• Italia ENEA Roma 1960: 2500 ITER Frascati
• Italia Eni 1953: 32000 + R&D 1500 + 5 raffinerie
• Italia Saipem 1957: 32000 EPC oil&gas multiphase
• Italia Snam 1941: 6000 30000 km gasdotti
• Italia De Nora 1923: 2000 electrochemistry green H2
• Italia Bracco Imaging: 3700 microbubble ultrasound
• Italia Chiesi Parma: 5500 respiratory pMDI DPI

Esempi numerici dettagliati

Esempio 1 - Water droplet 1 mm in air. rho_h=1000 (acqua), rho_l=1.2 (aria), L=0.001 m, sigma=0.0728. Eo = 998.8*9.81*(0.001)^2/0.0728 = 0.135. Surface tension dominated quasi-sferica + Laplace pressure 2*sigma/R. Italia: rain droplet + spray atomization injectors CNR-IRC Napoli combustion 200 + Chiesi Parma 5500 respiratory pMDI inhalers.

Esempio 2 - Air bubble 5 mm in water. L=0.005 m, water-air sigma=0.0728. Eo = 998.8*9.81*(0.005)^2/0.0728 = 3.36. Mixed regime ellipsoidal bubble. Italia: bubble columns industrial reactors Versalis Eni + Tecnimont 6000 EPC + Eni R&D San Donato 1500 + Politecnico Milano DEIB LCM.

Esempio 3 - Large bubble 50 mm in water. L=0.05 m. Eo = 998.8*9.81*(0.05)^2/0.0728 = 336. Strong gravity large flat-interface spherical-cap bubble Davies-Taylor terminal velocity. Italia: pipelines two-phase flow Snam 30000 km gasdotti + Saipem 32000 offshore platforms + Eni Mediterraneo + slug churn annular flow regime.

Esempio 4 - Mercury droplet 5 mm in air. rho_h=13534, rho_l=1.2, L=0.005, sigma=0.486. Eo = 13533*9.81*(0.005)^2/0.486 = 6.83. Mixed regime ellipsoidal mercurio dropping. Italia: laboratorio scientifico INFN Frascati + INRIM Torino metrologia + Politecnico Milano DEIB DENG.

Esempio 5 - Microfluidic 10 micron oil-water. rho_h=1000 (water), rho_l=800 (oil), L=1e-5 m, sigma=0.025. Eo = 200*9.81*(1e-5)^2/0.025 = 7.85e-6. Strong surface tension dominated spherical droplet T-junction microfluidic chip. Italia: Politecnico Milano DEIB LCM Laboratorio Microreattori + STMicroelectronics Catania MEMS + Sant'Anna Pisa biomechanics microfluidics 1987.

Limiti e considerazioni

1. Two-phase assumption: Eo defined heavy-light interface 2 fluids. Multi-phase 3+ phases requires multiple Eo per interface pair.
2. Static vs dynamic: Eo statico geometry-only; dynamics aggiungere Re Reynolds + Weber We + Capillary Ca + Morton Mo.
3. Surface tension temperature: sigma decrease con temperature. Eotvos rule sigma*V^(2/3) = K_E*(T_c-T). Use spot value at conditions.
4. Surface tension surfactants: surfactants reduce sigma 0.030 → 0.001 N/m factor 30. Eo varia massive.
5. Bubble shape regime Grace 1976: Eo-Re-Mo diagram defines spherical/ellipsoidal/cap bubble shapes industrial.
6. Capillary length L_c: scale separation Eo1 large drops gravity dominate. Transition L=L_c.
7. Wettability + contact angle: solid surface theta wetting affects droplet shape sessile/pendant adhesive.
8. Reduced gravity: micro-gravity space ISS Eo→0 surface tension dominate everywhere; lunar gravity 1.62 m/s^2 + Mars 3.71 m/s^2.
9. Non-Newtonian: polymer solutions + biological fluids non-Newtonian shear-thinning/thickening modify dynamics.
10. Pressure dependence: high pressure reduces sigma + alters delta_rho gas phase. CO2 supercritical sigma drop 0.005 N/m.

Errori comuni

Confondere Eo con Bo: stesso numero adimensionale. Eotvos number = Bond number = (rho_h-rho_l)*g*L^2/sigma. Convention chemical engineering Bo; geophysics fluid dynamics Eo.

L definizione: bubble diameter D vs radius R; capillary tube diameter D vs radius R. Convention specific application.

delta_rho heavy-light: rho_h fase pesante - rho_l fase leggera. Errore fase invertita -> Eo negativo o sbagliato sign.

Surface tension surfactants: surfactants riducono sigma drasticamente (factor 30 + clean -> contaminated). Use spot value at conditions.

Gravity reduced micro-g: ISS spacecraft + parabolic flights g~0.01-0.1 m/s^2 micro-gravity. Eo→0 surface tension dominate everywhere.

Applicazioni pratiche

1. Bubble columns reactors industrial: Italia Versalis Eni Mantova Ferrara Brindisi Priolo Gargallo + Tecnimont 6000 EPC + Eni R&D San Donato 1500 + Saipem 32000 EPC oil&gas + Ansaldo Energia 3000.
2. Multiphase flow pipelines: Snam 30000 km gasdotti + Saipem Castorone Castor 6 + offshore Mediterraneo + Eni Plenitude two-phase flow stratified slug churn annular regime.
3. Spray atomization combustion: CNR-IRC Napoli 200 droplet dynamics + Politecnico Milano DENG combustion + Sapienza Roma DIMA gas turbine.
4. Microfluidic + lab-on-chip droplet generation: Politecnico Milano DEIB LCM Laboratorio Microreattori + STMicroelectronics Catania MEMS + Sant'Anna Pisa biomechanics microfluidics 1987.
5. Pharmaceutical pMDI + DPI inhalers: Chiesi Parma 5500 + Boehringer Ingelheim Spiriva + GSK Ventolin Seretide + AstraZeneca Symbicort Turbuhaler respiratory aerosol drug delivery.
6. Microbubble ultrasound contrast agents: Bracco Imaging Milano 3700 + interventional radiology + medical imaging cardiovascular ecocontrastografia.
7. Boilers + steam generators: Ansaldo Energia 3000 + Eni R&D + ENEA Frascati nuclear research + Politecnico Milano DENG.
8. Electrochemistry chlor-alkali + green H2: De Nora 2000 + diaphragm membrane + green hydrogen electrolyzers PEM AEM AECN bubble dynamics gas evolution H2 O2.
9. Marine engineering + ship hydrodynamics: Fincantieri 22000 cantieri navali + CNR-INM Roma + Genova INSEAN bacino 470 m + Politecnico Milano DAER.
10. Universita ricerca chemical engineering + multiphase flow: Politecnico Milano DEIB + DENG + DCMC + LCM 7500 + Politecnico Torino DENERG 5000 + Federico II Napoli DICMaPI + Bologna DICAM + Padova DII + Pisa DICI + Politecnico Bari DIChim + Salerno DIIN + Genova DICCA + IIT 1700.

Aggiornato a luglio 2026