Faculteit Ingenieurswetenschappen
Vakgroep Chemische Proceskunde en Technische Chemie Laboratorium voor Petrochemische Techniek Directeur: Prof. Dr. Ir. G. B. Marin
Simulation of a Steam Cracker using Grey and Non-Grey Radiation Models
Wouter Foubert
Promoter: Prof. Dr. Ir. G. J. Heynderickx, Prof. Dr. Ir. G. B. Marin Coach: Ir. G. Stefanidis
Scriptie ingediend tot het behalen van de academische graad van burgerlijk scheikundig ingenieur
Academiejaar 2005 – 2006 Preface
Little in this work would have been possible without the support and advice of my coach George. I would like to take this opportunity to sincerely thank him and wish him the best, both professional and personal. Hopefully we’ll keep in touch.
Graag zou ik professor Heynderickx en professor Marin bedanken omdat ze mij de mogelijkheid geboden hebben om dit werk tot stand te brengen.
Het studentenleven is in vele opzichten een beetje een strijd, daarom een woordje van dank voor mijn strijdmakkers en, hopelijk, vrienden voor het leven Bart, Filip en Tom. Jullie zijn altijd welkom!
Ik zou ook graag al mijn medestudenten en dan vooral deze van de Sterre bedanken voor de leuke momenten en aangename babbels.
Astrid en Myriam, zonder jullie was alles anders verlopen. Heel erg bedankt voor jullie steun en advies.
Tenslotte een speciaal dankwoord voor mijn moeder, vader, broers en zus.
Wouter 13 juni 2006
FACULTEIT INGENIEURSWETENSCHAPPEN
Chemische Proceskunde en Technische Chemie Laboratorium voor Petrochemische Techniek Directeur: Prof. Dr. Ir. Guy B. Marin
Opleidingscommissie Scheikunde
Verklaring in verband met de toegankelijkheid van de scriptie
Ondergetekende, Wouter Foubert afgestudeerd aan de UGent in het academiejaar 2005 - 2006 en auteur van de scriptie met als titel:
Simulation of a Steam Cracker using Grey and Non-Grey Radiation Models
verklaart hierbij: 1. dat hij geopteerd heeft voor de hierna aangestipte mogelijkheid in verband met de consultatie van zijn/haar scriptie: o de scriptie mag steeds ter beschikking gesteld worden van elke aanvrager o de scriptie mag enkel ter beschikking gesteld worden met uitdrukkelijke, schriftelijke goedkeuring van de auteur of de promotoren o de scriptie mag ter beschikking gesteld worden van een aanvrager na een wachttijd van jaar o de scriptie mag nooit ter beschikking gesteld worden van een aanvrager
2. dat elke gebruiker te allen tijde gehouden is aan een correcte en volledige bronverwijzing
Gent, (datum)
(Handtekening)
Krijgslaan 281 S5, B-9000 Gent (Belgium) tel. +32 (0)9 264 45 16 • fax +32 (0)9 264 49 99 • GSM +32 (0)475 83 91 11 • e-mail: [email protected] http://www.tw12.ugent.be/ UNIVERSITEIT GENT
Faculteit Ingenieurswetenschappen Vakgroep Chemische Proceskunde en Technische Chemie Laboratorium voor Petrochemische Techniek Directeur: Prof. Dr. Ir. G. B. Marin
Simulation of a Steam Cracker using Grey and Non-Grey Radiation Models
Wouter Foubert
Promotor: Prof. Dr. Ir. G. J. Heynderickx, Prof. Dr. Ir. G. B. Marin Coach: Ir. G. Stefanidis
Academiejaar 2005-2006
Outline of the thesis
The first objective of this work is to make a comparative study between a grey and non-grey gas radiation model when applied in a simulation of an industrial steam cracking furnace segment with radiation burners. Results show a 6.3% difference in thermal efficiency, the grey gas model results being the highest. A second objective is a study on the influence of the emissivity of the furnace wall on the thermal efficiency of a steam cracking furnace. Therefore two coupled furnace-reactor simulations are performed with different wall emissivities. Results show an increase of thermal efficiency, naphtha conversion and ethylene yield when a high-emmisivity coating is applied to the wall.
In Chapter 1, an introduction on steam cracking and the simulation tools is presented. The objectives of the thesis are also defined. Chapter 2 gives an overview of the most common approximate radiative heat transfer models described in the literature. In Chapter 3 an overview is given of the most widely used models for estimation of radiative properties of gases. In Chapter 4 simulation results of grey and non-grey gas CFD simulations of an industrial naphta cracking furnace are presented. This is done by quantifying the differences in results of grey and non-grey simulations concerning important predicted variable profiles, like the flue gas flow and temperature profile, as well as the heat fluxes to the tubes. Chapter 5 investigates the influence of high-emissivity coatings in steam cracking furnaces. The efficiency of the application of high-emissivity coatings on the furnace walls in steam cracking technology can only be evaluated on the basis of a description of radiative heat transfer in frequency bands. To this end, a non-grey gas radiation model based on the Exponential Wide Band Model (EWBM) is developed and applied in the context of three- dimensional CFD simulations of an industrial naphtha cracking furnace with side-wall radiation burners. General conclusions are presented in Chapter 6. Simulation of a Steam Cracker using Grey and Non-Grey Radiation Models
Wouter Foubert Promotoren: Prof. Dr. Ir. G. J. Heynderickx, Prof. Dr. Ir. G. B. Marin
Abstract In this work, three-dimensional CFD radiation the latter is of paramount importance that it simulations of a steam cracking furnace with radiation is modelled in a sufficiently accurate manner. burners are performed. To execute these simulations the The first aim of this work is a comparative study commercial Computational Fluid Dynamics (CFD) between grey and non-grey radiation models applied software package FLUENT is used. The focus is on to an industrial scale steam cracking furnace. modelling of radiative heat transfer since this is the most important mode of heat transport in the radiative section Emission and absorption of radiation by gases show of a steam cracking furnace. While radiation property different behaviour from emission and absorption of variations for opaque solids are fairly smooth, gas radiation by surfaces. While the radiation property properties exhibit very irregular wavelength variations for opaque solids are fairly smooth, gas dependencies. As a result, two general types of models properties exhibit very irregular wavelength have been developed to describe the radiative properties dependencies. As a result, two general types of of a gas: grey and non-grey gas models. In this work, models have been developed to describe the radiative these two types of models are compared for a steam properties of a gas: grey and non-grey gas models. cracking furnace segment. Secondly, the non-grey gas Non-grey gas models take into account the wavelength radiation model is used in the context of coupled furnace- reactor simulations to investigate the effect of applying dependencies of the radiative properties of gases. high-emissivity wall coatings on the furnace thermal Most non-grey models, applicable for practical efficiency and the cracking results. simulations, divide the complete wavelength interval into distinct bands which are either absorbing or non- Keywords: Steam cracking, high emissivity coatings, absorbing and determine average values per band. In grey vs. non-grey radiation modelling. contrast to the non-grey models, grey gas models provide total radiative property values independent of the wavelength interval. Because this grey gas I. INTRODUCTION modelling requires only one solution of the Radiative Steam cracking of hydrocarbons is a petrochemical Transfer Equation (RTE) over the entire wavelength process in which saturated hydrocarbons are broken spectrum using an average absorption coefficient, it is down into smaller, often unsaturated, hydrocarbons. It by far the most widely used approach in practical is the principal industrial method for producing the radiative systems. The obvious drawback of the grey lighter alkenes (or commonly olefins), including model is the ill-correspondence between the model ethylene and propylene. The steam cracking process and the physical reality, i.e. the band-absorbing real takes place in reactor coils that are suspended in a gas compared to the grey gas. In this work a grey gas furnace with two main sections: a convection section and a non-grey gas radiation model are developed and and a radiant section. A hydrocarbon feed stream applied in the context of an industrial scale steam enters the furnace at the top of the convection section cracking furnace segment. Both are based on the and is initially heated by heat exchange with flue gas, Exponential Wide Band Model of Edwards. These mixed with steam and further heated to incipient models are implemented into FLUENT through the cracking temperature (500-680°C, depending on the use of User Defined Functions (UDF). The target is to feedstock). The stream then enters a fired tubular quantify the effect of the grey gas approximation on reactor (suspended in the radiant section or fire box) important predicted variable profiles like the one of where, under controlled residence time, temperature flue gas flow and temperature as well as on the reactor profile, and partial pressure, it is heated from 500- tube heat flux. These are important parameters for the 650°C to 750-875°C in 0.1-0.5s. During this short optimal design and operation of the furnace. reaction time hydrocarbons in the feedstock are The second aim of this work is to perform a cracked into smaller molecules. The firebox itself is coupled full furnace-reactor simulation. This is done heated by gas-fired wall-mounted radiant burners by linking FLUENT with the in-house reactor and/or floor mounted long-flame burners to a simulation program COILSIM. These coupled temperature of 1000°C to 1200°C. simulations which make use of a non-grey gas radiation model are used to investigate the influence of II. OBJECTIVES the furnace wall emissivity on the heat fluxes to the The research in this work will focus on modelling reactor tubes and on the thermal efficiency of the of radiative heat transfer in steam cracking furnaces. entire furnace. The fluxes determine significantly the Since, in the radiant section, over 90% of all heat cracking process inside the coils (e.g. feed stock transfer from the flue gas to the reactor coils is due to conversion and coke formation) and an improvement
I in thermal efficiency would mean less fuel is required emission into wall surface outgoing radiation inthe to achieve the desired reactor conversion. To assess clear bands. It was found that applying a high- the impact of the furnace wall emission coefficient two emissivity coating on the furnace wall increases the coupled furnace-reactor simulations are performed net outgoing radiation from the wall through the clear using a non-grey gas radiation model and two bands and decreases the net outgoing radiation from drastically different grey wall emissivities. In the first the wall through the absorption bands. Since radiation simulation the emission coefficient is set to a value of traveling through clear bands can reach the reactor 0.386 and in the second simulation the wall emissivity tubes without partially being absorbed by the flue gas, coefficient is set to 0.738. In both cases the wall is to as opposed to radiation traveling through absorption be considered a grey medium. Both of the wall grey bands, the thermal efficiency of the furnace increases. emissivities that are mentioned above are derived from non-grey wall emissivity valuesi. It is believed that IV. CONCLUSIONS increasing the wall emissivity will also increase the The quantitative analysis of the grey gas thermal efficiency of the entire furnacei,ii. This approximation shows a significant difference in the phenomenon is investigated in detail in this work in results between the grey and non-grey gas radiation order to identify the physical reasoning behind it. models. These differences should be taken into account in the process of the furnace design. III. RESULTS a) Grey vs. Non-grey gas modelling The results of coupled furnace-reactor simulations The target of this part is to quantify the effect of using CFD and a non-grey gas radiation model show the grey gas approximation that is commonly used for an increase of the furnace thermal efficiency when simulations of industrial applications on important increasing the furnace wall emissivity (applying a predicted variable profiles like the ones of the flue gas high-emissivity coating). An increase in thermal flow and temperature as well as reactor tube heat flux. efficiency can be translated into an increase in the These are important parameters for the optimal design products yields or a decrease in the required fuel input. and operation of the furnace. The comparison between the two models shows that when the grey gas i Heynderickx GJ, Nozawa M. Banded gas and non-gray surface simplification is used more energy is emitted by the radiation models for high-emissivity coatings. AIChE Journal. 2005;51(10):2721-2736. flue gas in the furnace box, and thus, more energy is ii Hellander JC. Throughput enhancement and tube temperature transferred to the process gas in the reactor tubes. stabilization. Hydrocarbon Processing. 1997;76:91-96 Thus, the predicted thermal efficiency increases from 37.6% when using the non-grey gas model to 43.9% when using the grey gas model. This 6.3% difference in the predicted thermal efficiency is quite large considering the scale and the importance of the industrial process and should be taken into account by the furnace designer. It has also been shown that although both models reproduce identical basic characteristics of the flow pattern in the furnace, noticeable quantitative differences in the flue gas speed are predicted in some regions of the furnace domain. b) High-emissivity coatings The efficiency of the application of high-emissivity coatings on the furnace walls in steam cracking technology can only be evaluated on the basis of a description of radiative heat transfer in frequency bands. To this end, a non-grey gas radiation model based on the Exponential Wide Band Model (EWBM) has been developed and applied in the context of three-dimensional CFD simulations of an industrial naphtha cracking furnace with side-wall radiation burners. The simulation results show that applying high emissivity coatings on the furnace walls improves the thermal efficiency of the furnace, and the cracking results. The increase in thermal efficiency should be attributed to the energy reallocation mechanism over clear and absorption bands taking place on the furnace walls. Wall surface incident radiation originating from gas absorption bands is partially converted due to wall
II Prefix
Table of contents
Preface ...... Overview...... Extended Abstract...... Table of contents...... List of Symbols...... Chapter 0. Nederlandstalige samenvatting ...... i 0.1. Inleiding...... i 0.2. Modelleren van stralingswarmteoverdracht...... iii 0.3. Modellen voor het schatten van stralingseigenschappen van gassen ...... v 0.4. Vergelijking van grijs gas en niet-grijs gas stralingsmodellen...... vii 0.5. Onderzoek van hoge emissiviteitscoatings in stoomkrakingsovens...... vii Chapter 1. Introduction...... 1 1.1. Overview...... 1 1.2. Furnace Design ...... 2 1.3. Simulation of a Steam Cracker ...... 3 1.4. CFD description...... 5 1.5. Objectives ...... 6 Chapter 2. Modelling of radiative heat transfer...... 8 2.1. Introduction...... 8 2.2. Radiative Heat Transfer...... 9
Table of contents p.1 Table of contents
2.3. Numerical methods...... 10 2.3.1. Differential Methods...... 10 2.3.1.a. Moment Method...... 11 2.3.1.b. Flux Model...... 12 2.3.1.c. Discrete Ordinates Method ...... 14 2.3.1.d. Finite Volume Method...... 17 2.3.2. Ray-tracing Methods...... 18 2.3.2.a. Semi-stochastic Monte Carlo Model...... 19 2.3.2.b. Discrete Transfer Model ...... 20 2.4. Conclusion ...... 21 Chapter 3. Models for estimation of radiative properties of gases ...... 23 3.1. Introduction...... 23 3.2. Radiative properties ...... 24 3.3. The Spectral Band models ...... 25 3.3.1. Narrow Band models ...... 25 3.3.2. Wide Band models...... 26 3.3.2.a. Exponential Wide Band Model...... 26 3.3.3. Correlated-k model ...... 32 3.4. Grey models...... 33 3.4.1. WSGG model...... 33 3.4.2. SLW model...... 33 3.4.3. Grey gas EWB modelling...... 34 3.4.3.a. Block Calculation Procedure ...... 34 3.4.3.b. Block Approximation method ...... 35 3.4.3.c. Band Energy Approximation ...... 35 3.5. Conclusion ...... 36 Chapter 4. Comparison of grey gas and non-grey gas radiation models ...... 39
Table of contents p.2 Table of contents
4.1. Introduction...... 39 4.2. Furnace segment geometry and operating conditions...... 40 4.3. Practical implementation of the radiation models ...... 42 4.3.1. Grey gas modelling...... 42 4.3.2. Non-grey gas modelling...... 43 4.4. CFD modelling approach...... 45 4.5. Results and discussion ...... 46 4.6. Conclusion ...... 51 Chapter 5. Investigation of high-emissivity coatings in steam cracking furnaces...... 53 5.1. Abstract...... 53 5.2. Introduction...... 53 5.3. Exponential Wide-Band Model (EWBM) ...... 54 5.3.1. Calculation of band transmittance ...... 54 5.3.1.a. Four region expression...... 56 5.3.1.b. Integration method...... 58 5.3.2. Non-grey gas modelling...... 59 5.4. Overview of the furnace/reactor calculations ...... 61 5.4.1. Reactor Model...... 61 5.4.2. Furnace Model ...... 62 5.4.2.a. Flow ...... 62 5.4.2.b. Radiation...... 62 5.4.3. Coupled furnace/reactor simulations ...... 64 5.5. Furnace geometry and operating conditions...... 65 5.6. Results and discussion ...... 68 5.7. Conclusions...... 76 Chapter 6. General conclusions ...... 78
Table of contents p.3 Prefix
List of Symbols
A* Dimensionless band absorption [-] A Band absorption or "effective bandwidth" [1/m] Surface ΔyΔz [m²]
Ai Coefficient of the Flux Model [W/m²] a Absorption Coefficient [-] a j Blackbody emission weighing factor [1/m]
B Surface ΔxΔz [m²]
Bi Coefficient of the Flux Model [W/m²] b Self-broadening to foreign-gas broadening ratio [-] b j Blackbody emission weighing factor [-]
c2 Second radiation constant, c2 = 0.014388 [m·K]
C Surface ΔxΔy [m²]
Ci Coefficient of the Flux Model [W/m²]
C1 , C2 , C3 Parameters in old EWBM c p, j Heat capacity of component j [kJ/kmole/K] d Spectral line spacing [1/m] d t Internal tube diameter [m]
E Total energy [m2/s2]
a List of Symbols r ei Coordinate direction i
F Surface [m²]
Fj Molar flow rate [Kmol/s] f (T /ν ) Fraction of the black body emissive power [-] f Fanning friction factor [-] f ()k Distribution function
G Incident radiation [W/m²]
3 Gb Generation of turbulent kinetic energy due to buoyancy [W/m ]
3 Gk Generation of turbulent kinetic energy due to mean velocity gradients [W/m ] g Statistical weighing factor for degeneracy g()k cumulative k-distribution function gr Gravitational acceleration [m/s2] h Specific enthalpy [J/kg]
I Radiation intensity [J/m2/s] r J Diffusion flux [Kg/m2] k Production rate of turbulent kinetic energy [m2/s2] Molecular conductivity [W/m/K] Absorption coefficient [1/m] kt Turbulent conductivity [W/m/K] keff Effective conductivity (k + kt ) [W/m/K]
L Mean beam length [m] M Number of directions Number of Bands n Empirical fudge factor [-] Refractive index [-]
b List of Symbols nr Normal pointing out of the domain
N r Number of reactions
Pe Equivalent broadening pressure parameter [-]
P,p Pressure [Pa]
Pa Absorber partial pressure [Pa]
P0 Reference partial pressure, 101325 Pa
Q,q Heat flux [W/m2, kW/m2]
3 qrad Volumetric heat release due to radiation [W/m ]
rk Rate of reaction k rb Radius of the bend [m] r Position vector sr Direction vector sr′ Skattering direction vector
S Line intensity [m/kg] Source term in conservation equation Thickness of gas layer [m] S /d Line intensity to line spacing ratio [m2/kg] T Temperature [K] t Time [s] u The quantity c2ν k /T [-]
Velocity of the process gas [m/s]
V Volume [m³] wm Weight [-] x Absorber mole fraction, x = Pa / P [-]
X Density path length [kg/m2]
c List of Symbols
X s Molar fraction of participating gas
Y a function z Axial coordinate [m] Band
Greek letters α Integrated band intensity [m/kg] absorptance [-] Band absorptance [-] Conversion factor depending on the units of p
β Mean line width to spacing parameter [-]
γ Line half width [1/m]
Parameter in Narrow Band Model
δ Unit tensor
δ k Change in vibrational quantum number of kth mode [-]
δ Parameter of Narrow Band Model Δ Difference operator [-] Δε Correction factor for overlapping [-]
ΔH k Heat of reaction k [kJ/kmol]
ε Emissivity [-] Dissipation rate of turbulent kinetic energy [m2/s3] ζ Nekrasov factor for bends [-]
η The quantity βPe [-]
y-direction cosine [m]
θ R Radiation Temperature [K] κ Absorption coefficient [1/m]
d List of Symbols
Parameter in Narrow Band Model λ Wavelength [m]
μ Molecular viscosity [kg/m/s]
x-direction cosine [m] ν Wavenumber [1/m]
ξ z-direction cosine [m]
ρ Gas density [kg/m3]
σ Stefan-Boltzmann constant, σ = 5.67 ⋅10−8 [J·s-1·m-2·K-4]
σ s Skattering coefficient [1/m]
σ k ,σ ε ,C1ε ,C2ε ,C3ε Model constants in the standard k − ε model [-]
τ H Maximum optical depth at the band head [-]
τ Transmittance [-]
τ Stress tensor [N/m2]
υ Vibrational quantum number [-]
υr Overall velocity vector [m/s]
υkj Stoichiometric coefficient of the component j in reaction k [-]
Φ Line-width-to-spacing temperature-variation parameter [-] Phase function [-] Ψ Band-intensity temperature-variation parameter [-] ω angle [-]
Exponential decay width [1/m] wetted parameter of the tube [m] Ω′ Solid angle [sr] Ω Hemispherical solid angle [sr] Cross section [m2]
e List of Symbols
Upper- and Subscripts 0 Reference value, origin b Black body c Center eff Effective (implies summation of molecular and turbulent properties) f Surface g Gas i Gas band Number x, y or z-direction j Gaseous species k Gas band Reaction l Lower m Direction
Number n Cell number
λ Spectral min minimum max maximum p Cell Centre pos Positive direction neg Negative direction ν Spectral t turbulent u Upper w Wall
f List of Symbols x x-direction y y-direction z z-direction
g Chapter 0
Nederlandstalige samenvatting
0.1. Inleiding Stoomkraken is een petrochemisch proces waarbij verzadigde koolwaterstoffen afgebroken worden in kleinere, meestal onverzadigde, koolwaterstoffen. Het is de belangrijkste industriële methode voor de productie van lichtere alkenen (olefines), zoals ethyleen en propyleen. Wereldwijd wordt op deze manier zo een 120 miljoen ton ethyleen per jaar (2004) geproduceerd, waarbij de grootste stoomkrakers een capaciteit hebben van ongeveer 2 miljoen ton per jaar. Bij stoomkraken wordt er een gas of vloeistofvormige KWS stroom, zoals nafta, LPG of ethaan samen met stoom kort verhit in een oven. Gezien de omzetting van verzadigde koolwaterstoffen in onverzadigde olefines sterk endoterm is, is een hoge warmtetoevoer vereist. Typisch is de reactietemperatuur zo’n 850°C en neemt de reactie slechts enkele fracties van een seconde in beslag. Nadat de krakingstemperatuur wordt bereikt, wordt het gas snel gekoeld in een “Transfer Line Exchanger”. Een schematisch overzicht van een stoomkrakingsoven kan gevonden worden in Chapter 1 Figuur 1-1. Een koolwaterstofstroom stroomt de oven bovenaan de convectiesectie binnen waar ze opgewarmd wordt door warmteuitwisseling met het verbrandingsgas en gemengd wordt met stoom. Vervolgens stroomt de voeding het stralingsgedeelte, “de firebox”, van de oven binnen. Hier vindt, onder gecontroleerde omstandigheden (temperatuur, druk en verblijftijd), het krakingsproces plaats. De “firebox” zelf wordt verwarmd met branders die zowel aan de wand (stralingsbranders) als in de vloer van de oven (lange vlam branders) kunnen geplaatst zijn. Na het verlaten van de oven wordt de productstroom snel afgekoeld en in een scheidingstrein opgesplitst in zijn verschillende componenten.
i Nederlandstalige samenvatting
Om competitief te kunnen zijn moet een stoomkrakingsinstallatie zeer flexibel zijn wat mogelijke voedingen betreft, ethaan tot gas-olie en zelfs niet-conventionele voedingen moeten kunnen verwerkt worden. De eerste computersimulatieprogramma’s waren dan ook vooral gericht op het voorspellen van de opbrengst bij verschillende voedingen. Het wordt echter snel duidelijk dat om goede voorspellingen van de opbrengst van een stoomkrakingsoven te kunnen maken de warmteoverdracht van oven naar reactorbuizen op een precieze manier moest kunnen gemodelleerd worden. Wat op zijn beurt dan weer een gedetailleerde kennis van de temperaturen en stroomprofielen vereist. Het is enkel sinds de introductie van “Computational Fluid Dynamics” (CFD) en de drastische verbetering van de computersnelheid dat het mogelijk wordt om zo’n gedetailleerd onderzoek van de “firebox” gekoppeld met opbrengstvoorspellingen in de reactorbuizen zelf, te kunnen uitvoeren. Het onderzoek in dit werk zal zich richten op het stralingsgedeelte van de krakingsoven en meerbepaald op het, via Computational Fluid Dynamics (CFD), modelleren van stralingswarmteoverdracht. Gezien, in de stralingssectie, meer dan 90% van alle warmteoverdracht door straling gebeurt is het van primordiaal belang dat deze op een voldoende correcte manier gemodelleerd wordt. Het eerste doel van deze thesis is het geven van een overzicht van de meest gebruikte modellen voor stralingsberekeningen. Eerst zal het modelleren van stralingswarmteoverdracht besproken worden, vervolgens zal de focus verschuiven naar modellen voor het schatten van stralingsparameters van gassen. Een tweede deel van dit werk zal bestaan uit een vergelijkende studie tussen grijze en niet-grijze stralingsmodellen toegepast op een industriële stroomkrakingsoven. Emissie en absorptie van straling bij gassen in het algemeen vertoont grote verschillen met de stralingseigenschappen van de meeste oppervlakken. Waar de stralingseigenschappen van ondoorzichtige vaste materialen meestal vrij continu variëren, zijn de stralingseigenschappen van gassen sterk golflengte afhankelijk. Vanuit dit inzicht zijn twee types modellen voor het modelleren van stralingseigenschappen van gassen gegroeid: grijze en niet-grijze modellen. Niet-grijze modellen houden rekening met de golflengte afhankelijkheid van gasstraling. De meeste niet-grijze modellen, beschikbaar voor praktische toepassingen, delen het gehele golflengte interval op in verschillende deelintervallen waar het gas ofwel absorberend of niet-absorberend kan zijn. Dit alles in contrast met de grijze modellen waar globale waarden voor de stralingseigenschappen
ii Nederlandstalige samenvatting worden gemodelleerd voor het gehele golflengte interval. Dit laatste heeft als voordeel dat er slechts eenmalig een stralingsoverdrachtsvergelijking moet opgelost worden en dit voor het gehele golflengteinterval. Bij de niet-grijze modellen daarentegen moet per golflengteinterval een afzonderlijke vergelijking opgelost worden, wat deze berekeningen een stuk computationeel complexer maken dan deze bij een grijs model. De grijze modellen zijn dan ook veruit de meest gebruikte modellen toegepast bij het simuleren van industriële toepassingen. Echter, het voordehandliggende nadeel van het gebruik van het grijze model is de slechte overeenkomst tussen het model en de fysische realiteit. In dit werk zullen zowel een grijs als een niet-grijs model ontwikkeld en toegepast (d.m.v. het commerciële CFD-pakket FLUENT) worden op een segment van een industriële stoomkrakingsoven. Beiden zullen gebasseerd zijn op het “Exponential Wide Band Model” (EWBM) van Edwards. Het doel is het inschatten van het effect van de grijze-gas-veronderstelling op belangrijke procesprofielen zoals de gasstroom- en temperatuursprofielen en de invloed op de warmtefluxen naar de reactorbuizen. Deze parameters zijn van belang bij het optimaal bedrijven en ontwerpen van een oven. Een derde en laatste deel van dit werk is een onderzoek naar de invloed van de overwand emissiviteit op de warmtefluxen naar de reactorbuizen en op de thermische efficiëntie van de gehele oven. Deze warmtefluxen beïnvloeden het krakingsproces in de buizen (bv. reactoropbrengst en cokesvorming) en een verbetering van de thermische efficiëntie betekent minder verbruik van brandstof voor eenzelfde reactoropbrengst. Om de invloed van de ovenwandemissiviteit in te schatten zullen twee afzonderlijke gecombineerde oven-reactor simulaties uitgevoerd worden. Bij beide simulaties zal een niet-grijs gasstralingsmodel en een grijs wandstralingsmodel vooropgesteld worden. Een eerste simulatie zal gebeuren met een emissiecoefficiënt van 0.386, een tweede met een emissiviteit van 0.736. Er wordt verondersteld dat het verhogen van de emissiviteit van de wand een positieve invloed zal hebben op de thermische efficiëntie van de oven. Het is het hoofddoel van dit derde deel om deze veronderstelling te controlleren en indien ze geldig blijkt een gedetailleerde kwalitatieve verklaring, gebaseerd op simulatieresultaten, te geven van dit fenomeen.
0.2. Modelleren van stralingswarmteoverdracht Zoals reeds eerder vermeld is stralingswarmteoverdracht de dominante vorm van warmtetransport in de stralingssectie van de stoomkrakingsoven. Dit maakt een accurate
iii Nederlandstalige samenvatting berekening van de stralingswarmteoverdracht in de ovenomgeving zeer belangrijk. De vergelijking die de stralingswarmteoverdracht beschrijft is van integro-differentiale vorm. Het is dan ook onmogelijk om deze op een analytische manier op te lossen. Verder bestaan er slechts enkele exacte numerieke oplossingmethoden, bijvoorbeeld de zone methode en de Monte Carlo methode. Deze methodes zijn echter computationeel zeer intensief en dus minder geschikt voor gebruik in praktische toepassingen. Voor toepassingen in simulatieprogramma’s waar tegelijkertijd berekeningen gebeuren op het gebied van stroming, chemische reacties en warmteoverdracht zijn er dan ook benaderende en snellere oplossingsmethoden ontwikkeld. De meest benaderende oplossingmethoden kunnen onderverdeeld worden in differentiële en “Ray- tracing” methoden: Differentiële methoden:
• Momenten methode
• Flux methode
• Discrete Ordinates (DO) methode
• Finite Volume (FV) methode Ray-Tracing methoden:
• Semi-stochastisch Monte Carlo model
• Discrete transfer model (DTM) De differentiële methoden transformeren de stralingswarmteoverdrachtsvergelijking in een set van partiële differentiaalvergelijkingen. Na discretisatie, bv. met het eindige-volume algoritme, kunnen deze vergelijkingen met behulp van snelle solvers opgelost worden. Bij de Ray-tracing methoden wordt een ééndimensionele vergelijking opgelost voor verschillende individuele stralen die gevolgd worden doorheen de gehele geometrie. Door het uitvoeren van deze procedure voor een veelvoud aan stralen kan een accuraat beeld van de stralingswarmteoverdracht opgemaakt worden. De belangrijkste overweging die in acht moet worden genomen bij het kiezen van een geschikt benaderend model voor stralingswarmteoverdracht is precisie t.o.v. rekentijd. Alhoewel modellen zoals het semi-stochastische Monte Carlo model, en in mindere mate het DO en FV model, de gebruiker in staat stellen om zeer precieze resultaten te verkrijgen, vereisen ze veel rekentijd vooral wanneer complexe geometriën beschouwd worden en wanneer er rekening moet
iv Nederlandstalige samenvatting gehouden worden met chemische reacties. Aan de andere kant, relatief simpele modellen zoals het momenten model of het Rossland model zijn snel, maar minder accuraat. Desalniettemin, ten gevolge van de voordurende verbeteringen in computertechnologie zijn modellen zoals het DO of FV model binnen het bereik gekomen van ingenieurstoepassingen zoals het simuleren van een industriële stoomkrakingsoven. Een tweede overweging is toepasbaarheid en algemeenheid. Sommige modellen, zoals het Moment, Rosseland en Flux model worden afgeraden in toepassingen waar het absorberende medium een lage optische dichtheid heeft. Terwijl anderen, zoals het DTM, ten gevolge van het Ray-Effect, minder geschikt zijn voor geometriën met een zeer fijn grid. Het is de opinie van de auteur van dit werk dat het DO-FV model, zoals ondersteund door FLUENT, een goede balans heeft tussen algemeenheid, vereiste rekentijd en accuraatheid. Het kan gebruikt worden in een brede waaier toepassingen van oppervlakte-naar-oppervlakte straling tot stralingberekingen bij verbrandingsproblemen. En alhoewel het meer rekenkracht vereist dan het DO-model, verzekert het conservatieve karakter van dit model accurate resultaten in meer complexe geometriën. Verder voorziet FLUENT bij dit model ook de aanzet tot niet-grijs modelleren. Deze argumenten maken het een bruikbaar model voor het bereiken van de twee hoofddoelen van dit project: het onderzoeken van de verschillen tussen grijs en niet-grijs modelleren en de invloed van de emissiviteit van de ovenwand op de thermische efficiëntie van de oven.
0.3. Modellen voor het schatten van stralingseigenschappen van gassen Emissie en absorptie bij gassen verloopt zeer verschillend van deze bij de meeste oppervlakten. Atomen en molecules in een verbrandingsgas rond 2500K vertonen overgangen tussen verschillende vibrationele en rotationele energietoestanden. Deze energietoestanden hebben een discreet karakter waardoor absorptie en emissie van stralingsenergie plaatsgrijpt in discrete spectrale lijnen bij specifieke golflengtes. Deze spectrale lijnen hebben in principe een infinitisimale dikte maar kunnen door verbredingseffecten overlappen en zogenaamde absorptie banden vormen. Er bestaan drie hoofdklassen van modellen voor het schatten van stralingsparameters van gassen. Ze worden hier neergeschreven in volgorde van dalende complexiteit:
• Spectrale lijn-na-lijn modellen. (SLBL)
v Nederlandstalige samenvatting
• Spectrale band methoden. (SB)
• Grijze modellen. Het SLBL model berekent de stralingsparameters voor elke individuele absorptielijn. Sinds rotationele en vibrationele banden verschillende duizenden absorptie lijnen bevatten, is deze methode computationeel zeer belastend en meer geschikt als benchmark voor andere modellen dan als echt praktisch toepasbaar model. Het SB model deelt het gehele spectrum op in verschillend golflengte intervallen. In deze intervallen wordt vervolgens een gemiddelde waarde voor elke stralingsparameter berekend. Afhankelijk van de spectrale resolutie wordt van Narrow Band (NB) of Wide Band (WB) modellen gesproken. In tegenstelling tot de bandmodellen beschouwen de grijze modellen gemiddelde waarden voor de stralingsparameters over het gehele golflengte interval. Deze worden berekend uit een gewogen gemiddelde van de spectrale of bandeigenschappen over het gehele golflengte interval. In praktische verbrandingssystemen is het begrenzen van de rekentijd één van de meest vooraanstaande bekomernissen. Deze systemen vereisen namelijk het modelleren van gelijktijdig optredende stroming, chemische reacties en warmteoverdracht. Gezien grijs modelleren het oplossen van slechts één stralingstransfervergelijking vereist over het gehele golflengte interval, is het veruit het meest gebruikte model in praktische verbrandingssystemen. Het voor de hand liggende nadeel van dit model is de slechte overeenstemming met de fysische realiteit. In dit werk zal de vergelijking tussen grijs en niet-grijs modelleren gebeuren aan de hand van het “Exponential Wide Band Model” van Edwards. Dit model wordt gekozen omdat het één van de meest gekende en gebruikte modellen is. Recent heeft een ander model, het zogenaamde Correlated-k (CK) model dat ontwikkelde wordt in het vakgebied van atmosferische fysica, opgang gemaakt in het vakgebied van hoge temperatuurverbranding. Een eerste verschil tussen beide modellen is dat het CK model de absorptiecoëfficiënt ontmiddellijk berekent terwijl het EWBM van Edwards eerst de effectieve bandbreedte en transmissiviteit bepaalt. Ten tweede, het CK model maakt gebruik van het idee dat de absorptiecoëfficiënt, welke sterk kan wijzigen binnen eenzelfde golflengte interval, herschikt kan worden volgens frequentie in een monotoon stijgende functie. Dit maakt spectrale integratie eenvoudig in tegenstelling tot de “four-region” uidrukking gebruikt bij het EWBM.
vi Nederlandstalige samenvatting
Er kan besloten worden dat het CK model enkele theoretische voordelen heeft t.o.v. het EWBM, alhoewel, tot op dit punt, geen echte vergelijkende studies in de literatuur beschreven staan.
0.4. Vergelijking van grijs gas en niet-grijs gas stralingsmodellen Het is één van de doelen van dit werk om de “grijs gas” simplificatie die toegepast wordt in veel praktische verbrandingsprocessimulaties te valideren. Hiervoor wordt een segment van een industriële stoomkrakingsoven gesimuleerd gebruikmakende van het DO-FV model aanwezig in FLUENT. Het EWBM, voor zowel het grijze als het niet-grijze model, wordt een afzonderlijke User-Defined-Function geprogrammeerd en vervolgens aan FLUENT gekoppeld. Om tot een duidelijk vergelijk van beide modellen te komen wordt de invloed van de grijze-gas-approximatie op verschillende belangrijke procesvariabelen, zoals temperatuurs-, stromings- en warmteflux profielen, nagegaan. De simulatieresultaten tonen dat wanneer de grijze-gas-simplificatie wordt toegepast er meer energie wordt geëmitteerd door het verbrandingsgas, en dus, dat er meer energie wordt overgedragen van het verbrandingsgas naar de reactorbuizen. De voorspelde thermische efficiëntie stijgt dan ook van 37.6%, wanneer het niet-grijze model wordt toegepast, naar 43.9% bij het grijze model. Dit verschil van 6.3% in thermische efficiëntie is vrij groot, zeker gezien de schaal en belangrijkheid van het industriële proces en moet dan ook in beschouwing genomen worden door ovenontwerpers. Er wordt ook aangetoond dat alhoewel beide modellen dezelfde stromingskaraketristieken voorspellen er verschillen in stromingssnelheden tussen beide modellen kunnen vastgesteld worden.
0.5. Onderzoek van hoge emissiviteitscoatings in stoomkrakingsovens Het effect van het toepassen van een hoge emissiviteitscoatings aan de ovenwand in een stoomkrakingsoven kan alleen geëvalueerd worden wanneer een niet-grijs stralingsmodel wordt toegepast voor het beschrijven van de stralingswarmteoverdracht. Daarom wordt een niet-grijs stralingsmodel gebaseerd op het “Exponential Wide Band Model” van Edwards ontwikkeld en toegepast in de context van een drie-dimensionele CFD simulatie van een industriële stoomkrakingsoven met stralingsbranders. De simulatieresultaten tonen dat het aanbrengen van een hoge emissiviteitscoating een gunstige invloed heeft op de thermische efficiëntie van de oven, de naftaconversie en de ethyleenproductie. Het stijgen van de thermische efficiëntie kan toegeschreven worden aan een herschikking van stralingsenergie die plaatsgrijpt aan de
vii Nederlandstalige samenvatting ovenwanden. Invallende stralingsenergie afkomstig van het verbrandingsgas en reizend door de absorptiebanden wordt gedeeltelijk omgevormd tot uitgaande stralingsenergie in de niet- absorberende/niet-emitterende banden. Er wordt vastgesteld dat het verhogen van de wandemissiviteit dit herschikkingseffect bevordert en dus dat de uitgaande stralingsenergie aan de wand zal stijgen in de niet-absorberende banden en zal dalen in de absorberende banden. Gezien stralingsenergie die reist door de niet-absorberende banden vrij de reactorbuizen kan bereiken, in tegenstelling tot de absorberende banden waar een vermindering in uitgaande stralingsenergie aan de wand “aangevuld” wordt door het emitterende verbrandingsgas, zal de thermische efficiëntie van de oven stijgen.
viii Chapter 1
General introduction
1.1. Overview Steam cracking is a petrochemical process in which saturated hydrocarbons are broken down into smaller, often unsaturated, hydrocarbons. It is the principal industrial method for producing the lighter alkenes (or commonly olefins), including ethylene and propylene. Some of the worlds largest crackers nowadays have an ethylene capacity of up to 2 million metric tons per year, compared to a world wide capacity of about 120 million metric tons per year (2004). In steam cracking, a gaseous or liquid hydrocarbon feed like naphtha, LPG or ethane is diluted with steam and then briefly heated in a furnace. Since the conversion of saturated hydrocarbons in the radiant tubes is highly endothermic, high energy input rates are needed. Typically, the reaction temperature is around 850°C but the reaction is only allowed to take place very briefly. In modern cracking furnaces, the residence time is reduced to fractions of a second, resulting in gas velocities reaching velocities up to and beyond the speed of sound, in order to improve the yield of desired products. After the cracking temperature has been reached, the gas is quickly quenched to stop the reactions in a transfer line exchanger. The products produced in the reaction depend on the composition of the feed, the hydrocarbon to steam ratio, on the cracking temperature and on the furnace residence time. Light hydrocarbon feeds, such as ethane, LPGs or light naphthas, give product streams rich in the lighter alkenes, including ethylene, propylene and butadiene. Heavier hydrocarbon feeds – full range and heavy naphthas as well as other refinery products – also give products rich in aromatic hydrocarbons and hydrocarbons suitable for inclusion in gasoline or fuel oil. The higher cracking temperature, also referred to as severity, favours the production of ethylene and benzene, whereas lower severity produces relatively higher amounts of propylene, C4-hydrocarbons and liquid products.
1 General introduction
The process also results in the slow deposition of coke on the reactor walls. This degrades the effectiveness of the reactor, therefore reaction conditions are designed to minimize this. Nonetheless, a steam cracking furnace can only run less than two months in-between two decoking cycles.
1.2. Furnace Design The principal arrangement of a cracking furnace is shown in Figure 1-1. A furnace can be split into three main sections: the convection section, the radiant section and the transfer line exchanger.
Figure 1-1: Principal arrangement of a cracking furnace1 A hydrocarbon feed stream enters the furnace at the top of the convection section and is subsequently heated due to heat exchange with the flue gas, mixed with steam and further heated to the incipient cracking temperature (500-680°C, depending on the feedstock). The stream then enters a fired tubular reactor (suspended in the radiant section or fire box) where, under controlled residence time, temperature profile, and partial pressure, it is heated from 500-650°C to 750-875°C in 0.1-0.5s. During this short reaction time hydrocarbons in the feedstock are cracked into smaller molecules. The firebox itself is heated by gas-fired wall-mounted radiant
2 General introduction burners and/or floor mounted long flame burners to a temperature of 1000°C to 1200°C. The reaction products leaving the radiant coil at 800-850°C are cooled in the transfer line exchanger to 550-650°C within 0.02-0.1s to prevent degradation of the highly reactive products by secondary reactions. The resulting product mixture is then separated into the desired products by using a complex sequence of separation and chemical-treatment steps.
1.3. Simulation of a Steam Cracker In order to be competitive a steam cracking plant needs to be flexible and able to process a variety of hydrocarbon feeds ranging from ethane to gas oils and even non-conventional feed stocks. For these reasons the initial efforts, in generating adequate simulation tools, contained the development of yield predicting reactor simulation software. Nevertheless, it was quickly realised that achieving adequate predictions of yields and conversions required the simulation software to be able to simulate the heat transfer from the furnace towards the reactor coils, which in turn required detailed knowledge of temperature and flow profiles in the furnace itself. This resulted in the introduction of combined furnace-reactor simulations. Since these combined simulations allowed insight in the mechanisms taking place in the radiation section of the furnace, this type of simulations made it also possible to predict run lengths in between decoking cycles, etc. With respect to the cracking furnace itself this resulted in the ability to precisely locate and assess heat release of the combustion gas and the interaction between flow and heat transfer to the coils. It was only until the introduction of Computational Fluid Dynamics (CFD) (see paragraph 1.4.) in chemical process engineering and the dramatic increase of computer power, available at an affordable cost, that researchers had the opportunity to perform such simulations in an elaborate way. A significant amount of work on developing a combined furnace-reactor simulation has been performed at the Laboratorium voor Petrochemische Techniek. For the simulation of the steam cracking process inside the reactor tubes a detailed kinetic radical reaction scheme was developed containing over 1000 reactions between 128 species. For reactor tubes with smooth internal surfaces under typical operating conditions, it is assumed satisfactory to combine the cracking kinetics with a one-dimensional plug-flow reactor model to achieve an acceptable degree of accuracy. More detailed information on this reactor simulation program, COILSIM, can be found in Willems and Froment2. The simulation of a combined furnace-reactor system requires
3 General introduction the calculation of the heat fluxes to the reactor tubes. For this purpose FURNACE was developed. This program calculates the heat fluxes according to the zone method of Hottel and Sarofim3. The furnace is divided into a number of surface and volume zones that are considered to be isothermal. For these zones the energy balances, containing radiative, convective and conductive contributions, are constructed. Solving these energy balances requires the assumption of a flow and heat release field in the furnace. The initial simulations were performed with an idealized flow field of the flue gas (such as plug flow) and more or less arbitrary chosen locations of heat release, especially when long flame burners are used, but the development of an in-house CFD code, FLOWSIM4, showed differences in results when more realistic flow and combustion patterns were assumed. Nowadays several commercial, off-the-shelf, CFD software packages are available. Although those programs have the disadvantage of being less flexible than in-house codes – the user doesn’t have access to the source code of the program – they have some advantages. They compensate for this lack of flexibility with a variety of built-in functionalities, and they also present the possibility of adding custom made codes, by use of User-Defined-Functions (UDF). Furthermore these CFD computer codes have an easy-to-use graphical user interface (GUI) for setting up problems and inspecting the results which makes them effective in tackling complicated (flow) geometries. Finally, the off-the-shelf codes have been optimized, by programming experts, on computational efficiency and stability which makes them generally more effective and stable than in-house made codes. In this work, the commercial CFD package FLUENT will be used for the simulation of the furnace with UDF’s appended if necessary. Where combined furnace-reactor simulations are required, FLUENT will be used in combination with the in-house code COILSIM. The main reason for choosing FLUENT is that it provides a basis for non-grey radiation modelling (see further), i.e. the possibility to divide the complete wavelength spectrum into distinct bands, which is currently not possible with the in-house CFD code FLOWSIM. One of the risks of computer modelling is that too much confidence might be given to the results. Some tend to believe anything generated by a computer. However, if the computer model has not been properly validated (e.g. by comparing results with experimental work or results obtained with other models), then the results may be unreliable. It is in the interest of the user to exercise a good judgment. Knowledge of the model’s capabilities helps to understand which results are
4 General introduction more reliable and which ones are less reliable. For example, in many cases, models are very useful in predicting trends, but are often very inaccurate in predicting the actual values. Since little experimental data are available on steam cracking furnaces the focus of this work will be more on comparison of results, taking into account the possibilities and limitations of the used models, than on quantitatively interpreting calculated data.
1.4. CFD description In essence, Computational Fluid Dynamics (CFD) is the use of computers to analyze problems in fluid dynamics. All flowing fluids obey the three laws of conservation: conservation of mass, momentum and energy. This makes it possible to describe their behaviour by means of mathematical equations. Since analytical solution of these equations is impossible, CFD attemps to solve the governing equations of flow – often in combination with models for radiation, chemical reactions and mechanical movement – numerically. This makes it possible to study processes where experimental work is very expensive, difficult or even impossible to perform. Although there are various CFD approaches, all follow the same methodology: 1. The geometry (physical bounds) of the problem is defined. 2. The volume occupied by the fluid is divided into discrete cells (the mesh). 3. The physical models are set, i.e. the governing equations of flow, often in combination with other models (e.g. radiation, reaction, mechanical movement), are defined. 4. Boundary conditions are defined. This involves specifying the fluid behaviour and properties at the boundaries of the problem. For transient problems, the initial conditions are also defined. 5. The numerical procedures are specified. This involves setting up the discretization schemes for the different terms (e.g. different upwind schemes for the convection operator) and choosing the solution algorithm (e.g. coupled vs. segregated solver and implicit vs. explicit treatment). 6. Analysis and visualization of the resulting solution (post processing).
5 General introduction
1.5. Objectives The research in this work will focus on modelling of radiative heat transfer in steam cracking furnaces. Since, in the radiant section, over 90% of all heat transfer from the flue gas to the reactor coils is due to radiation it is of paramount importance that radiation is modelled in a sufficiently accurate manner. A first aim of this work is to present the reader with an overview of the most widely used models in radiation calculations. First, modelling of radiative heat transfer will be discussed; secondly, the focus will shift to models for the estimation of radiative properties of gases. Efforts will be made to broaden this literature review from merely a listing up of models to a justification of the model-choices that have been made for practical use further on in this work. A second goal is a comparative study between grey and non-grey radiation models applied on an industrial scale steam cracking furnace. Emission and absorption of radiation by gases shows different behaviour from emission and absorption of radiation by surfaces. While the radiation property variations for opaque solids are fairly smooth, gas properties exhibit very irregular wavelength dependencies. As a result, two general types of models have been developed to describe the radiative properties of a gas: grey and non-grey gas models. Non-grey gas models take into account the wavelength dependencies of the radiative properties of gases. Most non- grey models, applicable for practical simulations, divide the complete wavelength interval into distinct bands which are either absorbing or non-absorbing and determine average values per band. In contrast to the non-grey models, grey gas models provide total radiative property values independent of the wavelength interval. Because this grey gas modelling requires only one solution of the Radiative Transfer Equation (RTE) (see Chapter 2) over the entire wavelength spectrum using an average absorption coefficient, it is by far the most widely used approach in practical combustion systems. The obvious drawback of the grey model is the ill-correspondence between the model and the physical reality, i.e. the band-absorbing real gas compared to the grey gas. In this work a grey gas and a non-grey gas radiation model are developed and applied in the context of an industrial scale steam cracking furnace segment. Both are based on the Exponential Wide Band Model of Edwards (see Chapter 3). The target is to quantify the effect of the grey gas approximation on important predicted variables like the flue gas flow and temperature profile as well as on the heat flux to the reactor coils. These are important parameters for the optimal design and operation of the furnace.
6 General introduction
A third and final aim of this work is to perform a full coupled furnace-reactor simulation. This is done by linking FLUENT with the in-house reactor simulation program COILSIM. The coupled simulation is used to investigate the influence of the furnace wall emissivity on the heat fluxes to the reactor tubes and on the thermal efficiency of the entire furnace. The fluxes significantly determine the cracking process inside the coils (e.g. feed conversion and cokes formation) and an improvement in thermal efficiency would imply that less fuel is required to achieve the desired feed conversion. To assess the impact of the furnace wall emission coefficient two coupled furnace-reactor simulations are performed using a non-grey gas radiation model and two drastically different grey wall emissivities. In the first simulation the emission coefficient is set to a value of 0.386 and in the second simulation the wall emissivity coefficient is set aqual to 0.738. In both cases the wall is to be considered a grey medium. Both of the grey wall emissivities that are mentioned above are derived from non-grey wall emissivity values5. It is believed that increasing the wall emissivity will also increase the thermal efficiency of the entire furnace5,6. The main goal of this third part is to investigate this phenomenon in detail and identify the physical reasoning behind it.
References
1 Ullman’s Encyclopedia of Industrial Chemistry
2 Willems P, Froment GF. Kinetic Modeling of the Thermal Cracking of Hydrocarbons, Part 1: Calculation of Frequency Factors, and Part 2: Calculation of Activation Energies. Industrial and Engineering Chemisty Research. 1988;27(11):1959-1971
3 Hottel HC, Sarofim AF. Radiative Heat Transfer. McGraw-Hill, New York. 1967
4 Prins A. Simulation of Industrial Steam Cracking Furnaces using Computational Fluid Dynamics. Proefschrift tot het verkrijgen van de graad van Doctor in de toegepaste wetenschappen: Scheikunde. Universiteit Gent. Vakgroep Chemische Proceskunde en Technische Chemie. 2005
5 Heynderickx GJ, Nozawa M. Banded gas and non-gray surface radiation models for high-emissivity coatings. AIChE Journal. 2005;51(10):2721-2736.
6 Hellander JC. Throughput enhancement and tube temperature stabilization. Hydrocarbon Processing. 1997;76:91- 96
7 Chapter 2
Modelling of radiative heat transfer
2.1. Introduction Radiative heat transfer is the most important mode of heat transport in the radiative section of a steam cracking furnace. This makes an accurate calculation of the radiative heat fluxes in the furnace domain of paramount importance. The governing equation of radiative heat transfer is of an integro-differential type. It is impossible to solve this equation in an analytical manner. Only a few numerical exact solution procedures exist to solve this equation, e.g. the zone method and the Monte Carlo method. These methods are excessively computer-time demanding and therefore currently not suitable for an incorporation in a simulation program for the calculation of fluid flow, chemical reactions and heat transfer. Approximate and rapid solution methods combining accuracy and computational economy are preferred especially for three-dimensional simulations. In this chapter an overview is given of the most common approximate radiative heat transfer models described in the literature. These can be classified into two categories: Differential and Ray-Tracing methods. Differential Methods:
• Moment Method
• Flux Model
• Discrete Ordinates Method
• Finite Volume Method Ray-Tracing Models
• Semi-stochastic Monte Carlo Model
• Discrete Transfer Model
8 Modelling of radiative heat transfer
2.2. Radiative Heat Transfer In contrast to the convective and diffusive heat transfer phenomena, thermal radiation is an electromagnetic phenomenon. In a physical space discretised by the finite volume method, this means that each volume is not only interacting with its direct neighbours but with all visible elements. The Radiation Transfer Equation (RTF) for an absorbing, emitting and scattering medium at position r in the direction sr is:
4 r r 4π dI()r, s r r 2 σT σ s r r r r + ()()a + σ s I r, s = an + I()()r, s´ Φ s ⋅ s´ dΩ (2-1) ds π 4π ∫0 Where: r the position vector, sr the direction vector, sr´ the scattering direction vector, s the path length, a the absorption coefficient, n the refractive index, σ s the scattering coefficient, σ the Stefan-Boltzmann constant, I the radiation intensity – depending on position and direction –, T the local temperature, Φ the phase function en Ω the solid angle. Figure 2-1 is a schematic visualization of the RTE (equation (2-1)).
Figure 2-1: Radiative Heat Transfer Neglecting refractive and scattering effects – acceptable in case of a steam cracking furnace with radiation burners – and assuming a grey medium gives the simplest form of the radiation transfer equation:
r r 4 dI()r, s ⎛σT r r ⎞ = a⎜ − I()r, s ⎟ (2-2) ds ⎝ π ⎠
9 Modelling of radiative heat transfer
The left hand side of equation (2-2) describes the change of radiation intensity I at the location r along the path length ds in the direction of the vector sr . This intensity is changed by the terms on the right hand side of equation (2-2) representing an increase through emission (see Figure 2-1) and a decrease due to absorption (see Figure 2-1). The principal intention of solving the radiation transfer equation is not to compute the intensity distribution in the considered domain but a source term Sradiation for the energy conservation equation.
⎛ ⎞ S = a⎜ IdΩ − 4σT 4 ⎟V radiation ⎜ ∫ ⎟ (2-3) ⎝ 4π ⎠
Equation (2-3) results from integrating the radiation transfer equation over all solid angles dΩ. As mentioned before, these equations are impossible to solve in an analytical manner. Rather, several numerical solution methods with different degree of accuracy are used to produce an expression for Sradiation.
2.3. Numerical Methods In general there are two different kinds of approximate radiation models: ray-tracing and differential methods. The essential difference of these two groups is the number of directions in which the transport equations are formulated.
2.3.1. Differential Methods The differential methods are tools to transform the equation of radiation transfer into a set of partial differential equations which can be formulated in the coordinate system of the fluid flow equations:
∂I ∂I ∂I μ +η + ξ = a()I − I ∂x ∂y ∂z b (2-4)
Where μ, η and ξ are the direction cosines in the x-, y- and z-direction of a Cartesian coordinate system. After discretising this equation with the finite-volume method fast solvers can be used to compute the radiation intensity I.
10 Modelling of radiative heat transfer
2.3.1.a. Moment Method
The moment method is based on the Milne-Eddington approximation to compute the radiation transfer in which a Taylor series expansion, truncated after the terms of first order, is used for the intensity: r r r r r r I()r, s = I 0 (r )()+ μI x r +ηI y (r )+ ξI z (r ) (2-5) By substituting equation (2-5) into the radiation transfer equation (2-4) a partial differential equation is found1:
∂ ⎛ 1 ∂I 0 ⎞ ∂ ⎛ 1 ∂I 0 ⎞ ∂ ⎛ 1 ∂I 0 ⎞ ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ = 3a()I 0 − I b (2-6) ∂x ⎝ a ∂x ⎠ ∂y ⎝ a ∂y ⎠ ∂z ⎝ a ∂z ⎠ The source term of the energy conservation can be computed from the intensity distribution as follows:
⎛ ⎞ S = a⎜ IdΩ − 4σT 4 ⎟V = 4aV πI − I radiation ⎜ ∫ ⎟ ()0 b (2-7) ⎝ 4π ⎠
The conservation of radiation energy applied to a wall gives:
Qt = −Qe + Qa (2-8)
In which Qt is the net radiative flux towards and normal to the surface, Qe is the flux emitted by the wall and Qa is the flux absorbed by the wall. The quantities Qt, Qe and Qa are given by:
Q = I sr nr ⋅ srdΩ t ∫ () 4π 4 Qe = ε wσTw (2-9) Q = α I sr nr ⋅ srdΩ a w ∫ () nr⋅sr<0
A partial differential equation of first order for the intensity I0 can be derived by substituting equations (2-9) into (2-8), integrating over the solid angle using (2-5) and assuming that Kirchoff’s law is valid:
2()2 − ε w ∂I 0 σ 4 ε w I 0 ± = ε w Tw (2-10) 3a ∂xi π
The signs (-) and (+) correspond to the positive and negative coordinate directions respectively.
11 Modelling of radiative heat transfer
An identical set of equations for intensity, boundary conditions and radiation source-term found for the moment method can be derived by the spherical harmonics method applying the P1- approximation and the Marshak boundary conditions2. The RTE (2-6) is a diffusion equation, which is easy to solve with little CPU demand, but it fails to register inhomogeneities of the intensity distribution since there is only one equation for the complete solid angle. H. Knaus, et al.3 report that the formulation of the boundary conditions for the intensities perpendicular to the wall surfaces only can cause overprediction of heat fluxes. The Fluent manual4 warns for a loss of accuracy, depending on the complexity of the geometry, if the optical thickness is small and for a possible overprediction of radiative fluxes from localized heat sources or heat sinks. Considering the relative simplicity, thus low computational cost, of the model it can be easily applied to complicated geometries. A simplified variant of the moment method, also supported by Fluent4, is the Rosseland Radiation Model5. The Rosseland model assumes that the intensity is a black-body intensity at the gas temperature, contrary to the moment method where the actual intensity is calculated through a transport equation (2-6). Since it does not solve an extra transport equation for the incident radiation (the intensity is directly linked with the gas temperature), the Rosseland model is faster than the moment method and requires less memory. But since it uses a black-body intensity as prediction for the local intensity it can only be used for optically thick media.
2.3.1.b. Flux Model
Several flux models exist, but the most widely used is the so-called six-flux model. This flux model is based on a Schuster-Schwarzschild approximation and was proposed by De Marco and Lockwood6. Similar to the moment method the intensity is represented by a truncated Taylor series expansion for the intensity in which the first and part of the second order terms are retained:
r r r r r 2 r 2 r 2 r I()r, s = μAx ()r +ηAy ()r + ξAz (r )+ μ Bx (r )+η By (r )()+ ξ Bz r (2-11)
The coefficients Ai and Bi are related to the intensities in the directions of coordinate axes,
pos neg pos neg I i − I i I i + I i Ai = Bi = i = x, y, z (2-12) 2 2
12 Modelling of radiative heat transfer
pos neg In which I i and I i represent the intensities in the positive and negative directions of the subscripted coordinate. The intensity distribution equation (2-11) is substituted into the transfer equation (2-4) and some mathematical operations have to be done (De Marco and Lockwood6) to yield three coupled second order equations for the quantities Cx, Cy and Cz:
∂ ⎛ 1 ∂C ⎞ ⎜ x ⎟ ∂x a ∂x ⎝ ⎠ 4 3 −1 −1 C x σT ∂ ⎛ 1 ∂C ⎞ ⎜ y ⎟ 4 ⎜ ⎟ = a −1 3 −1 C y − aσT ∂y ⎝ a ∂y ⎠ 4 −1 −1 3 C z σT ∂ ⎛ 1 ∂C ⎞ ⎜ z ⎟ ∂z ⎝ a ∂z ⎠ (2-13) π ⎛ 1 1 ⎞ C x = ⎜ Bx + By + Bz ⎟ 2 ⎝ 2 2 ⎠ π ⎛ 1 1 ⎞ C y = ⎜ Bx + By + Bz ⎟ 2 ⎝ 2 2 ⎠ π ⎛ 1 1 ⎞ C z = ⎜ Bx + By + Bz ⎟ 2 ⎝ 2 2 ⎠
Each heat flux Ci is dependent on the intensities in all six directions and therefore this model is named a coupled flux model. In simpler flux models this coupling is not guarantueed leading to non-physical coupling of heat flux and coordinate directions. The source-term for the energy conservation may be written as:
z ⎛ 1 4 ⎞ S radiation = 4aV ⎜ ∑Ci − σT ⎟ (2-14) ⎝ 3 i=x ⎠
The boundary conditions are determined as for the moment method and can be formulated as follows: 2()2 − ε ∂C ε C ± w x = ε σT 4 w x 3a ∂x w w
2()2 − ε ∂C y ε C ± w = ε σT 4 w y 3a ∂y w w (2-15) 2()2 − ε ∂C ε C ± w z = ε σT 4 w z 3a ∂z w w This six-flux method has a lot of similarities with the moment method; both have an equivalent set of equations for intensity, boundary conditions and radiation source-terms. Compared to only
13 Modelling of radiative heat transfer one RTE for the moment method, the flux method has three (2-13) which makes the model more sensitive for inhomogeneities of the intensity distribution, but increases the computational time. As for the moment method H. Knaus, et al3 report that the formulation of the boundary conditions for the intensities perpendicular to the wall surfaces only, can cause over prediction of heat fluxes. In general the model provides good predictions in all circumstances, except when significant heat release and low optical thickness are combined6. By increasing the number of terms retained in the Taylor’s series, correspondingly more approximated equations can be derived and the accuracy of the method is correspondingly enhanced. However, increasing the number of equations also increases the computer time needed for calculation.
2.3.1.c. Discrete Ordinates Method
In the discrete ordinates method7 (DO) the radiative transport equation is solved for a set of m discrete directions (m=1,2,…,M). In this way the integral over the solid angle/direction is r replaced by numerical quadrature weights wm summed over each ordinate. Each direction sm is associated with a solid angle in which the intensity is assumed to be constant. All solid angles are non-overlapping and spanning the total angle range of 4π. The discrete ordinates representation of the RTE for an absorbing-emitting, non-scattering, gray medium in a rectangular coordinate system can be written as:
∂I ∂I ∂I μ m +η m + ξ m = a()I − I m ∂x m ∂y m ∂z b m (2-16)
Where I m []≡ I()x, y, z;μ m ,η m ,ξ m is the total radiation intensity at position (x,y,z) in the discrete r direction sm (μm = cosθ, ηm = sinθ sinφ, ξm = sinθ cosφ). The finite-difference form of equation (2-16) can be obtained by multiplying equation (2-16) by dxdydz and integrating over a Cartesian control volumea as follows:
xd xu yd yu zd zu p μ m A(I m − I m )+η m B(I m − I m )+ ξ mC(I m − I m ) = aV (I b − I m ) (2-17)
p xd xu yd yu zd zu Where I m is the cell-centre intensity and I m , I m , I m , I m , I m , I m are the intensities at the faces of
xd yd zd the control volume. If the upstream face intensities I m , I m and I m are assumed to be known
a For reasons of simplicity a Cartesian mesh will be assumed, opposed to the tetragonal mesh used in the simulations.
14 Modelling of radiative heat transfer from boundary conditions for a control volume adjacent to a boundary of the enclosure, the following relationships8 can be used to eliminate the unknown downstream face intensities
xu yu zu I m , I m and I m in equation (2-17):
p xu xd I m = αI m + (1−α )I m p yu yd I m = αI m + ()1−α I m (2-18) p zu zd I m = αI m + ()1− α I m
The finite-difference weighing factor α = 0.5 represents the second-order diamond difference scheme proposed by Carlson and Lathrop9. After some rearrangement the cell-center
p intensity, I m , may be evaluated as:
xd yd zd p μ m AI m +η m BI m + ξ mCI m + α aVI b I m = (2-19) μ m A +η m B + ξ mC + α aV
Where
A = ΔyΔz,B = ΔxΔz, C = ΔxΔy (2-20)
p Every I m value on every cell-centre and face of the control volume can be computed by stepping from control volume to control volume. The direction of recursive evaluation is in accordance r with the direction of physical propagation of the radiation beam as defined by sm .
As mentioned before there is no solid-angle/direction integration, the integrals over solid angle, required to calculate quantities of interest such as radiative flux and energy source term
p distributions, are estimated by quadrature using the I m values. The quadrature weights, wm, are chosen to make specified integrals exact when the intensity is uniform; the usual integrals are10:
M dω = w = 4π ∫ ∑ m 4π m=1 M srdω = w sr = 0 ∫ ∑ m m (2-21) 4π m=1 M 4π sr ⋅ srdω = w sr ⋅ sr = δ ∫ ∑ m m m 4π m=1 3
Where δ is the unit tensor.
The weight wm can be thought of as a solid angle associated with direction m, but the exact boundaries of the solid angle are not usually defined geometrically. Because of the interest in
15 Modelling of radiative heat transfer calculating energy flows across boundaries that have normals aligned with the three coordinate r 10 directions ei , it is desirable to also satisfy the half-moments :
M sr ⋅ er dω = w sr ⋅ er ,0 = π i = 1,2,3 ∫ i ∑ m ()m i max (2-22) r r m=1 s⋅ei ≥0
The m directions are chosen to satisfy symmetry conditions such that, for example, identical results are obtained with any 90° rotation of the axes. For a total number of directions, M = 24,
48, 80, etc., all these conditions can be satisfied. For M = 8 (the S2 method), the symmetry property must be abandoned to permit (2-22) to be satisfied. The source-term of the energy conservation equation can now be computed from the difference of the intensities in all directions multiplied by the weights wm at the cell centres and the emissivity:
⎛ ⎞ ⎛ M ⎞ S = aV ⎜ IdΩ − 4σT 4 ⎟ = aV ⎜ w I p − 4σT 4 ⎟ radiation ⎜ ∫ ⎟ ∑ m m (2-23) ⎝ 4π ⎠ ⎝ m=1 ⎠
By analogy to the moment method and the flux model, the intensities at the boundaries result from the conservation of energy at the diffuse emitting wall:
σ 4 1 r r I m = ε w Tw + ()1− ε w ∑ wm' I m' n ⋅ sm' r r (2-24) π π n⋅sm' <0
The DO method is a generally applicable and widely used method. However, in some cases it fails to conserve radiation energy10. For applications of engineering interest, it happens rarely that the wall normals are directed along the Cartesian axes. Hence the equality in equation (2-22) will be violated (the summation will not equal π). This inconsistency may violate conservation at the wall. It may even cause a net flux to be calculated at the wall when intensities, incident to and leaving the wall, are uniform and equal. The discrete ordinates method poses also problems in handling specular reflection at walls and in handling anisotropic scattering10. But, these effects are negligible in case of steam cracking furnaces.
16 Modelling of radiative heat transfer
2.3.1.d. Finite Volume Method
In order to overcome the conservation problems of the Discrete Ordinates radiation model a conservative variant, called the Finite-Volume scheme (FV) and implemented in the Fluent software package, has been proposed. The finite volume method was originally introduced by Raithby and Chui11 and is slightly modified and described in detail by Baek, S.W. et al12 and Raithby10. By analogy to the discrete ordinates method the radiative transport equation is solved r for a set of M discrete directions sm (m = 1,2,…,M) and the intensities are assumed to be constant over one solid angle Ω m . The intensities at the cell centres are calculated exclusively as a function of their upstream neighbouring cells. However there are two important differences between these methods. The first difference is that for the finite volume method, not the numerical quadrature is used to compute the solid angle integrals, but the integration is done exactly. The set of differential r radiative heat transfer equations for the direction sm are solved by integrating over the control volume V and the solid angle Ω m to yield the balance of radiant energy:
p p ∑Qm, f = a(I b − I m )ΔΩ mV f (2-25)
Qm,f is the radiant energy within ΔΩ m that crosses a surface Ff of the control volume:
Q = I F ()sr ⋅ nr dΩ m, f m, f f ∫ f (2-26) ΔΩm
The second difference is that the determination of the intensity at the surface Im,f is done for the finite volume method by integrating the one-dimensional transport equation along a distance Δs and not through a simple interpolation method as described by equation (2-18):
−aΔs −aΔs 1 ⎛ ∂I b ⎞ −aΔs I m, f = I m,uf e + I b,m, f ()1− e − ⎜ ⎟ ()1− e ()1+ aΔs (2-27) a ⎝ ∂s ⎠ f
The intensity Im,uf at a point uf at a distance Δs upstream from the point on the surface f can be
p found by an interpolation of the intensities from the upstream neighbouring cell Im,nb and I m . Thus the intensity at the control volume can be expressed as a function of its upstream values. After the intensities have been determined the source-term of the energy conservation equation can be computed at a cell centre from the difference of the intensities and the emissivity:
17 Modelling of radiative heat transfer
⎛ ⎞ ⎛ M ⎞ S = aV ⎜ IdΩ − 4σT 4 ⎟ = aV ⎜ I p ΔΩ − 4σT 4 ⎟ radiation ⎜ ∫ ⎟ ∑ m m (2-28) ⎝ 4π ⎠ ⎝ m=1 ⎠
The boundary conditions for the intensities are calculated by the incoming radiant flux:
σ 4 ()1− ε w 1 I m = ε w Tw + ∑Qm' (2-29) π Fw π m'
The FV method is fully conservative. Analytical integration ensures exact satisfaction of all full moment conditions (e.g. equations (2-21)). Furthermore there is no loss of scattered radiation. The method allows the angular grid to be adapted to resolve sharp changes in the intensity with direction. There is no need to preserve symmetry with respect to 90° rotation, although symmetry will result if the angular grid is constrained to be symmetric. The FV method is also fully conservative at the boundaries of the domain, provided the standard rules for application of the FV method are obeyed. The radiation energy balance at the boundary is therefore exactly satisfied even when there is specular reflection. In principle, the discrete representation of the RTE compromises the accuracy of the solution. However, errors due to directional discretization (distinct form violation of conservation) reduce quadratically with refinement in the solid-angle grid. This error is particularly important at boundaries when solid angles are bisected by the boundary. It is important in these cases to correctly account for the overhanging fraction. This is done through the use of pixilation4. Overall the DO-FV model spans the entire range of optical thicknesses, and allows solving problems ranging from surface-to-surface radiation to participating radiation in combustion problems. It also allows the solution of radiation in semi-transparant media. Computational cost is moderate for typical angular discretizations, and memory requirements are modest. This makes it a very powerful model for simulating industrial applications.
2.3.2. Ray-tracing Methods For ray-tracing radiation models one-dimensional equations along a multitude of individual rays through the furnace chamber are solved. The formulation of the equations is made along straight, arbitrarily oriented rays, which inevitably are not aligned with the coordinate system of the fluid flow:
18 Modelling of radiative heat transfer
dI = a()I − I ds b (2-30) Determining all traces of the rays and cells located along their way requires either a lot of computational time or a lot of memory. In addition, the vectorisation and parallelisation of the algorithms of ray-tracing radiation models are limited due to the different lengths of the individual rays.
2.3.2.a. Semi-stochastic Monte Carlo Model
In the pure version of the Monte Carlo model the coordinates of the starting points as well as the directions of the rays are generated by random numbers in the whole computational domain. For the semi-stochastic Monte Carlo Model a fixed number of rays is started in the geometrical centre of the cells using a fixed angular distribution13. This can be done if the cells are small enough to assume that radiation is emitted almost isotropically over the solid angle. Each ray is traced from its starting point until it leaves the computational domain or to its extinction. The absorbed energy per cell is computed by summing over all rays:
−aΔs Qabs = ∑()Qn − Qn−1 = ∑Qn (1− e ) rays rays (2-31)
The source-term of the energy conservation equation of one cell is the difference between absorbed and emitted energy:
4 S radiation = Qabs − 4σ aT V (2-32) The reflection of rays at the wall in a direction computed by random numbers follows the usual Monte Carlo approach for an ideal diffusive reflector. From the incident energy flux of a ray the wall absorbs the amount of energy Qabs,w = ε wQn and the rest of the energy is reflected according toQn+1 = ()1− ε w Qn .
Since the Monte Carlo model needs a lot of computational time it is not recommended to use this method for simulations of full scale industrial applications. However, the high accuracy allows the use of the solution of this model as a benchmark/validation for other radiation models.
19 Modelling of radiative heat transfer
2.3.2.b. Discrete Transfer Model
The discrete transfer model14 (DTM) is built on the concept of solving the radiation transfer equation for representative rays in the furnace and to this extent it is related to the Monte Carlo model. But in contrast to the Monte Carlo model the directions of the rays are specified and not randomly chosen. The rays are started on the boundary surface only and are solved along paths between two boundary walls only and are not individually reflected at the walls and traced to extinction. An equation for the intensity results from the integration of the radiation equation along a distance in the direction of the ray:
−aΔs −aΔs I n+1 = I n e + I b (1− e ) (2-33) The intensity balance per cell and ray is given in the following equation:
−aΔs −aΔs I netto = I n − I n+1 = I n (1− e )− I b (1− e ) (2-34) The radiation source-term is the sum of all intensity changes caused by all rays passing through a cell.
S radiation = ∑()I netto ΔΩF rays (2-35)
For all rays starting at the walls of the computational domain only an intensity at the boundary is defined
⎛ ΔΩ ⎞ σ 4 I 0 = ()1− ε w ∑⎜ I nr⋅sr<0 ⎟ + ε w Tw (2-36) rays ⎝ π ⎠ π and therefore these rays are dependent on the intensities towards the boundary surface. This explicit treatment of the intensities at walls leads to a slightly slower convergence in comparison to the Monte Carlo model3. The advantage of the discrete transfer model is threefold: it is a relatively simple model, the accuracy can be increased by increasing the number of rays and it applies to a wide range of optical thicknesses. Since rays only start on the boundaries of the computational domain the computational time is reduced in comparison to the Monte Carlo Model. However because the illumination of the furnace is not guarantueed, the occurrence of the so-called ray effect can strongly affect the accuracy of the solution. Because information can only be spread along the traces of rays and only a fixed number of directions are defined, cells which are not affected by a particular ray can not get any information from it. This ray effect can cause an irregular heat flux
20 Modelling of radiative heat transfer and source-term distribution in the computational domain. The ray effect was first mentioned by Lathrop15 in connection with the discrete ordinates method. However, the sensitivity of the discrete ordinates method seems to be weaker than for the ray tracing models using the same number of rays3.
2.4. Conclusion The main consideration that has to be taken into account when choosing an appropriate model for radiative heat transfer calculation is accuracy vs. computational time. Though models like the semi-stochastic Monte Carlo model, and to a lesser extend the DO and FV models, yield accurate results, they are time consuming especially in simulations with complicated geometries, where predictions of fluid flow, chemical reactions and heat transfer have to be made. On the other hand, relative simple methods like the moment method or Rosseland model are quick but less accurate. Nevertheless due to constant improvements in computer technology the DO and FV models have come within reach of engineering applications like the simulation of a steam cracking furnace. A second consideration is applicability and generality. Some models, like the moment, Rosseland and flux model, should not be used in simulations where the absorbing medium has a low optical thickness, while others like the DTM are, due to the ray-effect, less fit for geometries with a very fine grid. It is the opinion of the author that the DO-FV method, as supported by FLUENT, has a good balance between generality, computational cost and accuracy. It can be used over a wide range of applications from surface-to-surface radiation to strong participating radiation in combustion problems, and, although it is more time consuming than the DO model, its conservative character guarantuees more accurate results in more complex geometries. Moreover it also supports the possibility of non-grey modelling (see further). These arguments make it a useful model to achieve the two main goals of this work: Investigating the differences between grey and non-grey modelling and the influence of the wall emissivity on the thermal efficiency of the furnace.
21 Modelling of radiative heat transfer
References
1 Özisik MN. Radiative Transfer and Interactions with Conduction and Convection. John Wiley & Sons, New York, 1973, Chap. 9, pp. 343-346
2 Koch R. Berechnung des mehrdimensionalen spektralen Strahlungswärmeaustauschs in Gastrubinen- Brennkammern: Entwicklung und Überprüfung von grundlagenorientierten Ansätzen und Methoden. Diss. 1992, Univeristät Karlsruhe, Inst. f. Thermische Strömungsmaschinen.
3 H. Knaus, R. Schneider, X. Han, J. Ströhle, U.Shell, K.R.G. Hein. Comparison of different radiative heat transfer models and their applicability to Coal-Fired Utility Boiler Simulations. University of Stuttgart, Institute for Process Engineering and Power Plant Technology.
4 Fluent Inc. Radiative Heat Transfer. Chapter 11.3, 2003.
5 R. Siegel and J. R. Howell. Thermal Radiation Heat Transfer. Hemisphere Publishing Corporation, Washington D.C., 1992.
6 De Marco AG, Lockwood FC, A new flux model for the calculation of radiation in furnaces. La rivista die combustibili;29(5-6);184-196, 1975
7 Selçuk N., Kayakol N. Evaluation of discrete ordinates method for radiative transfer in rectangular furnaces. International Journal for Heat and Mass Transfer. 1997;40(2):213-222
8 Fiveland WA. Three-dimensional radiative heat transfer solutions by discrete ordinates method. Journal of Thermophysics. 1988;2;309-316
9 Carlson BG, Lathrop KD. Transport theory – the method of discrete ordinates, in Computing Methods in Reactor Physics (Edited by H Greenspan, C.N Kelber and D.Orkent). Gordon & Breach, New York. 1968 pp 165-266
10 Raithby GD. Discussion of the finite-volume method for radiation, and its application using 3D unstructured meshes. Numerical Heat Transfer, 1999. Part B, 35:389-405
11 Raithby GD, Chui EH. A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. Journal of Heat Transfer. 1990;112:415-423
12 Baek SW, Kim MY, Kim JS. Nonorthogonal finite-volume solutions of radiative heat transfer in a three- dimensional enclosure. Numerical Heat Transfer, Part B. 1998;34:419-437
13 Richter W, Heap M. A Semistochastic Method for the Prediction of Radiative Heat Transfer in Combustion Chambers, Western States Section/The Comb. Inst., 1981 Spring Meeting, paper 81-17
14 Lockwood FC, Shah NG. A new radiation solution method for incorporation in general combustion prediction procedures. 18th Symp. (Int.) on Comb., The Comb. Inst.,1980, pp 1405-1414
15 Lathrop KD. Ray Effects in Discrete Ordinates Equations. Nuclear Science and Engineering, 1968, vol. 32, pp.357-369
22 Chapter 3
Models for estimation of radiative properties of gases
3.1. Introduction In this work, the focus is on modelling of radiative heat transfer in steam cracking furnaces. In chapter 1 an overview has been given on how to model radiative heat transfer, this chapter will focus on the estimation of radiative properties of gases such as the absorption coefficient, the absorptance and the transmissivity. The atoms and molecules in a typical flue gas in industrial combustion with temperatures up to 2500K exhibit transitions between bound energy modes associated with vibrational and rotational energy modes. This as opposed to bound-free and free-free transitions associated with dissociation, electron transitions and ionization, which in general only become important at much higher temperatures in the order of 10000 K. The bound-free and free-free transitions would give rise to a continuous contribution of the absorption and emission spectrum of the gas. The bound energy modes have a discrete character and as a result the emission and absorption of radiation energy originating from transitions between bound energy modes can only occur in so-called spectral lines at discrete wavelengths, corresponding to the energy differences between the bound energy modes (ΔE = hν). The spectral lines are in principle of infinitesimal width, but are broadened as a result of a number of effects. These effects include, among others, natural broadening as a result of the natural uncertainty in the exact position of the discrete energy levels and collision broadening determined by the collision rate between the atoms and molecules of the gas. These broadening effects cause the different spectral lines to overlap and merge into so- called absorption or emission bands. Three main classes of models can be identified. They are presented here in order of decreasing complexity.
23 Models for estimation of radiative properties of gases
• The Spectral Line-By-Line (SLBL) models.
• The Spectral Band (SB) models.
• The Grey models. The SLBL models calculate the radiative properties for each individual line (e.g. Hartmann, Di Leon and Taine)1 and they usually rely on the HITRAN database2. Since a rotational and vibrational band encompasses thousands of absorption lines, this approach is far too expensive from the computational point of view for engineering applications. At present, SLBL models are used only for benchmark solutions to approximate methods and as such they will not be further discussed in this chapter. The SB models divide the whole wavelength spectrum (~ 0.7 - 25 μm, where the vibration- rotation transitions in gases occur) into a number of absorbing/emitting spectral intervals (bands) over which the radiative properties are averaged. Depending on the spectral resolution, the SB models can be classified into Narrow Band (NB) models and Wide Band (WB) models. In contrast to the band models mentioned above, grey gas models provide total radiative property values, which are derived through weighted sum of spectral or band properties over the entire wavelength spectrum In this chapter an overview will be given of the most widely used spectral band and grey models. The exponential wide-band model (EWBM)3 as proposed by Edward will be discussed in more detail since it is the method used during simulations.
3.2. Radiative properties Solving the RTE’s requires an expression for the absorption coefficient a (see Chapter 2). On the other hand most models for estimating radiative properties of gases provide an expression for the transmissivity τ or absorptance α of a gas along a distance L (path length). In this paragraph an expression is derived for the absorption coefficient in function of the transmissivity or absorptance. The change in radiation intensity at a specific wavelength as a result of attenuation in a non- scattering medium depends on the magnitude of the local intensity.
dI λ = −aλ ()T, p, L I λ dL (3-1)
24 Models for estimation of radiative properties of gases
In which aλ, the absorption coefficient at wavelength λ, is also function of temperature, pressure and composition of the gas medium. Integration of the equation then yields Bourguer’s law for a non-scattering medium:
⎡ S ⎤ I L = I exp − a T, p, L* dL* λ () λ,0 ⎢ ∫ λ ()⎥ (3-2) ⎣ 0 ⎦
In the special case of a gas at uniform temperature, pressure and composition over the path length L, equation (3-2) can be written as:
I λ ()L = I λ,0 exp (− aλ L ) (3-3)
a Taking into account the definition for transmissivity I λ (L)()= τ λ L I λ ,0 or absorptance
I λ ()L = []1− α λ ()L I λ ,0 this results in
1 1 a = − ln[]τ ()L = − ln[]1−α ()L λ L λ L λ (3-4) To compute the local absorption coefficient at a control volume of a computational grid, some authors take L as the mean effective path length of that control volume and others take L as the mean beam length for the whole enclosure. However, there is no sound theoretical foundations for both approaches4. Both options are available in FLUENT.
3.3. The Spectral Band models As mentioned before the Spectral Band (SB) models can be classified into Narrow Band (NB) models and Wide Band (WB) models. Another band based method that has gained some popularity over the last years is the k-distribution method5 and its extension to non-homogeneous media, the so-called correlated k-method6,7.
3.3.1. Narrow Band models In the narrow band models the spectrum is divided into narrow wavenumber intervals, in which the spectral lines are assumed to have random spacing and/or intensity. The most popular NB models are the Elsasser8 model and the several statistical models9,10. The statistical narrow band
a Reflectivity ρ in the gas is assumed zero, α + λ + ρ = 1
25 Models for estimation of radiative properties of gases model of Malkmus10 is considered the most accurate narrow band model11. According to this model, the mean transmissivity τΔν over a wavenumber interval Δν, for a homogeneous and isothermal gas layer of thickness S at pressure p, is given by
⎡ γ ⎛ δ ⎞⎤ τ = exp⎢− 2 ⎜ 1+ X pSκ −1⎟⎥ Δν δ ⎜ s γ ⎟ (3-5) ⎣⎢ ⎝ ⎠⎦⎥
Where κ, 1δ and γ are parameters of the model and Xs is the molar fraction of the participating gas. For each of these wavenumber intervals a separate radiation transfer equation (RTE) is solved. The NB models yield very accurate solutions for high temperature gases and although they are computationally more efficient than the SLBL method they still require a large number of bands that render them very expensive for computation. Considering the accuracy of this model it is often used as a basis for comparison of other models.
3.3.2. Wide Band models The wide band models make use of the fact that even within a wide spectral interval radiation intensity can be approximated as being constant. In principle, WB correlations are derived by integrating NB results across an entire band and still provide sufficient accuracy. The Exponential Wide Band Model (EWBM)3 is by far the most celebrated model of this category.
3.3.2.a. Exponential Wide Band Model
The exponential wide band model3 is based on the fact that absorption and emission of infrared radiation from gases is generally concentrated within several bands resulting from changes of energy storage of the molecules between vibrational modes. Energy changes between rotational modes lead to a large number of spectral lines within each band. A detailed knowledge of the position, shape and intensity of these lines is considered to be unimportant in the EWBM, instead the band shape is approximated by an exponential function. The radiative properties are obtained by specifying three model parameters: the exponential decay width ω, the mean line width to
∞ spacing parameter β = πγ / d and the integrated intensityα = S d d ν −ν . Hence the mean 0 ∫ ()()0 0
−(ν −ν ) ω line intensity to spectral line spacing ratio S d equalsα ω e 0 . Three band shapes can be allowed in the exponential band model to better represent actual gas absorption bands:
26 Models for estimation of radiative properties of gases
• In case of an asymmetric band with the upper limit νu specified
−()ν u −ν ω S d = ()α ω e (3-6)
• In case of an asymmetric band with the lower limit νl specified
−()ν −ν l ω S d = ()α ω e (3-7)
• In case of an asymmetric band with the band centre νc specified
−2 ν −ν ω S d = ()α ω e c (3-8)
Theoretically α, β and ω vary with temperature and pressure within each band. The EWBM model however assumes variation only with respect to temperature. The pressure dependence is taken into account through the Pe number.
n ⎧ p ⎫ Pe = ⎨ []1+ ()b −1 x ⎬ (3-9) ⎩ p0 ⎭
Expressions for the functional dependence of α, β and ω have been developed elsewhere3,12 and are summarized below.
⎡ ⎛ m ⎞⎤ ⎢1− exp⎜− ∑ ± ukδ k ⎟⎥Ψ()T ⎣ ⎝ k =1 ⎠⎦ α()T = α 0 ()T (3-10) ⎡ ⎛ m ⎞⎤ ⎢1− exp⎜− ∑ ± u0,kδ k ⎟⎥Ψ()T0 ⎣ ⎝ k =1 ⎠⎦
m ∞ ⎡()υ k + g k + δ k −1 ! ⎤ ⎢ exp()− ukυ k ⎥ ∏ ∑ g −1 !υ ! k =1 υk =υ0,k ⎣ ()k k ⎦ Ψ()T = m ∞ (3-11) ⎡()υ k + g k −1 ! ⎤ ⎢ exp()− ukυ k ⎥ ∏ ∑ g −1 !υ ! k =1 υk =0⎣ ()k k ⎦
− 1 ⎛ T ⎞ 2 ⎛ Φ()T ⎞ ⎜ ⎟ ⎜ ⎟ β = β 0 ⎜ ⎟ ⎜ ⎟ (3-12) ⎝ T0 ⎠ ⎝ Φ()T0 ⎠
1 2 ⎧ m ∞ 2 ⎫ ⎪ ⎡()υ k + g k + δ k −1 ! ⎤ ⎪ ⎨ ⎢ exp()− ukυ k ⎥ ⎬ ∏ ∑ g −1 !υ ! ⎪ k =1 υk =υ0,k ⎣ ()k k ⎦ ⎪ Φ()T = ⎩ ⎭ (3-13) m ∞ ⎡()υ k + g k + δ k −1 ! ⎤ ⎢ exp()− ukυ k ⎥ ∏ ∑ g −1 !υ ! k =1 υk =υ0,k ⎣ ()k k ⎦
27 Models for estimation of radiative properties of gases
1 ⎛ T ⎞ 2 ⎜ ⎟ ω = ω0 ⎜ ⎟ (3-14) ⎝ T0 ⎠
Where uk = c2ν k /T and u0,k = c2ν k /T0 .
The subscript 0 denotes to a reference condition. T0 = 100 K is considered for convenience.
υ0,k denotes the lowest possible initial state and is taken equal to zero when δ k is positive or zero, and equal to the absolute value of δ k when δ k is negative. The set quantum numbers
()υ1 ,υ 2 ,...,υ k ,...υm describes the molecular, vibrational state before absorption of a photon of radiation energy.
Table 3-1: Exponential Wide Band Model Parameters12
Pressure parameters Spectral location Band absorption parameters Vibrations Bands Gas nb νl νc νu α0 β0 ω0 ν (cm-1) δ1,δ2,... k -1 -1 -1 -1 -2 -1 (T0 = 100 K) (cm )(cm)(cm)(cm/gm m )(cm)
m=3 (1) Rotational 1/2 b b 1 8.6(T0/T) +0.5 05200.0 0.14311 28.4 ν1=3652 0,0,0
ν2=1595 (2) 6.3 μm 1/2 1 8.6(T0/T) +0.5 1600 41.2 0.09427 56.4 ν3=3756 0,1,0
g1=1 (3) 2.7 μm g =1 (1) H2O 2 0,2,01/2 0.19 1 8.6(T0/T) +0.5 3760 0.13219 g3=1 1,0,0 2.3 60 0,0,1 22.4 (4) 1.87 μm 1 8.6(T /T)1/2+0.5 5350 3 0.08169 43.1 0,1,1 0 (5) 1.38 μm 1 8.6(T /T)1/2+0.5 7250 2.50.11628 32 1,0,1 0 m=3 (1) 15 μm 0.71.3 667 19 0.06157 12.7 ν1=1351 0,1,0
ν2=667 (2) 10.4 μm 0.8 1.3 960 2.47 x 10-9 0.04017 13.4 ν3=2396 -1,0,1
g1=1 (3) 9.4 μm -9 c a c 0.8 1.3 1060 2.48 x 10 0.11888 10.1 (2) CO2 g2=2 0,-2,1 g =1 (4) 4.3 μm 3 0.8 1.3 2410 110 0.24723 11.2 0,0,1 (5) 2.7 μm 0.65 1.3 3660 4 0.13341 23.5 1,0,1 (6) 2.0 μm 0.65 1.3 5200 0.066 0.39305 34.5 2,0,1 a The 1,0,0 band of the linear CO2 molecule appears only at high pressures when a dipole is induced by collisions. b -1/2 For the rotational band of H2O α(T) = α0 and β(T)=β0(T/T0) . c -1 Because of Fermi resonance between the ν1 and 2ν2 levels, the Ψ and Φ functions for the 1060 cm band are to be those of the 960 cm-1 band; i.e. use the set of δ’s for the 960 cm-1 band to get Ψ and Φ for either band.
28 Models for estimation of radiative properties of gases
After absorption, a set of quantum numbers (υ1 + δ1 ,υ2 + δ 2 ,...,υk + δ k ,...υm + δ m ) describes the th th transition creating the j absorption band for the i species. gk are statistical weights for generative and degenerative vibrations. ν 1 ,ν 2 ,...,ν k ,...ν m are wavenumbers associated with the characteristic molecule vibrations. Values for the exponential wide band model parameters participating in equations (3-9) to (3-14) 12 can be found in Table 3-1 for two species (H2O and CO2) and all absorption bands. Originally, Edwards3 provided correlations, derived from experimental data, for the calculation of the mean band transmittance. Therefore, best agreement with data is obtained when the data correlation values for α, β and ω are used together with this set of so-called four-region expression to obtain the band absorption or “effective” bandwidth A. A can be interpreted as the width of a black band centred about the middle of the real absorption band, that produces the same absorption as the real band does. The four-region expression consists of relations of linear, square root and logarithmic form with respect to the optical depth at the band head which depends on the level of absorption strength of the band.
• The linear region: τ H ≤ 1,τ H ≤ η
* A = τ H (3-15)
• The square root region η ≤ τ H ≤ 1 η ,η ≤ 1
* 1 2 A = ()4ητ H −η (3-16)
• The log-root region: 1 η ≤ τ H ≤ ∞,η ≤ 1
* A = ln()τ Hη + 2 −η (3-17)
• The logarithmic region: τ H ≥ 1,η ≥ 1
* A = ln()τ H +1 (3-18)
* Where A = A ω is the dimensionless band absorption, τ H = αX ω is the maximum optical depth at the band head andη = βPe .
The band transmittance is calculated from
29 Models for estimation of radiative properties of gases
* τ H dA τ k = * (3-19) A dτ H and the bandwidth from
A Δν = (3-20) ()1−τ k
Use of equation (3-19) implies a grey gas assumption for each band that breaks down at small path lengths. For that reason Edwards3 suggests an upper limit for the calculation of the band transmittance:
τ k = min()τ k ,0.9 (3-21) Imposing an upper limit for the band transmittance may introduce serious mistakes in case that a recursive relationship is used due to the strong dependence on the grid resolution13. One way to mitigate the problem is to define the band limits from the average properties of the domain in a pre-processing step. Then the band transmittance can be calculated from:
1− A
τ k = (3-22) Δν fix
Using the latter approach, the band absorption A is still calculated from the four-region expression (3-15) to (3-18) and the temperature and composition dependence are still taken into account. However, instead of calculating the band transmittance from equation (3-19) in a one step, the bandwidth Δν fix is computed from the average properties of the domain as a first step and equation (3-22) is used in a second step. Besides, the four-region expression, a second integration method was proposed by Cumber, Fairweather and Ledin14 for calculation of the mean band transmittance. They suggest that in order to avoid the grey gas assumption, which is necessary in order to calculate the band transmittance equation (3-19), one could calculate the spectral transmittance employing an analytical expression according to the exponential-tailed band model as:
30 Models for estimation of radiative properties of gases
⎡ ⎤ ⎢ ⎛ S ⎞ ⎥ − ⎜ ⎟ X j ⎢ ⎝ d ⎠i ⎥ τν = exp⎢∑ ⎥, X = ρL (3-23) ⎢ ij ⎛ S ⎞ ⎛ X ⎞ ⎥ 1+ ⎜ ⎟ ⎜ ⎟ ⎢ d ⎜ η ⎟ ⎥ ⎣ ⎝ ⎠i ⎝ ⎠ j ⎦ where i denotes the band and j the participating species. (S d )i values are calculated from relations (3-6) to (3-8). Summation over i is necessary when two or more bands of the same absorbing component overlap. Next, the average band transmittance is computed via integration of the spectral transmittance over the band wavelength spectrum:
1 τ = τ dν k ∫ ν (3-24) Δν Δν
The selection of the band limits in equation (3-24) does not significantly influence the results15.
In turn, the spectral absorption aν can be calculated from (3-4). For very small path lengths L, ()S d ()X η << 1, equation (3-23) can be simplified into:
⎡ ⎛ S ⎞ ⎤ τν = exp⎢∑ − ⎜ ⎟ X j ⎥ (3-25) ⎣ ij ⎝ d ⎠i ⎦ Applying (3-4) gives:
⎛ S ⎞ aν = ∑⎜ ⎟ ρ j (3-26) ij ⎝ d ⎠i
This last relation shows that for very small path lengths grid, independent solutions can be obtained by using this approach. On the other hand, a problem arising when employing this model is that the EWBM parameters given in Edwards (see Table 3-1) were derived from experimental data in conjunction with the four-region expression. When these parameters are used for numerical quadrature of equation (3-24) discrepancies in the calculated band absorption values obtained with the four-region expression method and the integration method are observed. The problem can be alleviated by adjusting ω upward by 20% when spectral integration is performed, as suggested by Edwards and Balakrishnan12 and by Edwards3. It is noted that adjustment of the ω value should be applied outside the linear region namely,τ H ≤ η ≤ 1.
31 Models for estimation of radiative properties of gases
3.3.3. Correlated-k model The correlated-k (ck) approach6,7 divides the spectrum into bands sufficiently narrow to assume the Planck function as constant in each band. The model makes use of the idea that the absorption coefficient field, which can considerably vary even within very small wavelength intervals, can be reordered in frequency so that it forms a monotonously increasing function. Then, the integration over wavenumber can be replaced by integration over absorption coefficient, which yields smoother functions to integrate, and can be approximated by an N-point quadrature.
A function Y dependent on aν, such as, for example, the spectral transmissivity τν, is averaged as follows7:
1 ∞ 1 Y a = Y a dν = Y k f k dk = Y k g dg ()ν ∫ (ν )∫ ()()∫ () () (3-27) Δν Δν 0 0 where f(k) is the distribution function of aν and g(k) is the cumulative k-distribution function given by
k g()k = ∫ f (k' )dk' (3-28) 0
Non-uniform media equation (3-27) relies on the scaling approximation, i.e., the spectral and spatial dependence of the absorption coefficient are separable. As mentioned before, the reordered absorption coefficient k(g) within a band is a smooth, monotonically increasing function of g, which can be easily calculated from high-resolution spectra using LBL calculation. Therefore, the average value of function Y may be accurately computed using a Gaussian type quadrature:
1 N Y a = Y k g dg ≈ ω Y g ()ν ∫ ()() ∑ j ()j (3-29) 0 j=1 where gj and ωj are the quadrature points and weights respectively and N is the quadrature order. For example, the transmissivity of a gas layer of length L averaged over a band of width Δν is calculated as (see paragraph 3.2)
N τ Δν = ∑ω j exp()− k j L (3-30) j=1
32 Models for estimation of radiative properties of gases
16,17 The absorption coefficients kj are given by
* X s pT k j k j = (3-31) TQ()T where Xs is the molar fraction of the absorbing species, Q(T) is the partition function of an
* isolated molecule and k j are parameters of the model. In a mixture of two absorbing gases, the transmissivity is taken as the product of the transmissivities of the individual species, and a double summation appears in equation (3-30).
3.4. Grey models As mentioned in the introduction, grey gas models, provide total radiative property values. These are derived through weighed sum of spectral or band properties over the entire wavelength spectrum. Two well known models of this category are the Weighted Sum of Grey Gases (WSGG) model18 and the Spectral Line-based Weighted sum of grey gases (SLW)19. A third approach is to calculate total radiation properties in the context of grey gas EWB modelling.
3.4.1. WSGG model The weighed-sum-of-grey-gases WSGG model18 expresses the total gas emissivity ε as a weighed sum of grey gas emissivities:
N g ε = ∑b j ()T []1− exp()− a j L (3-32) j=0 where bj and aj are the blackbody emission weighing factor and the absorption coefficient for the th j grey gas component respectively, and L is the path length. The coefficients bj and aj are obtained from a fit to total emissivity with the constraint that the sum of coefficients aj is equal to unity. The transparent regions of the spectrum are accounted for by the term j = 0. Assuming Kirchoff’s law is valid, equation (3-4) can be used to calculate the absorption coefficient of the medium.
3.4.2. SLW model The spectral line-based weighted-sum-of-grey-gases (SLW) has been developed by Denison and Webb19. It expresses the total emissivity of the gas by an expression identical to that used in the
33 Models for estimation of radiative properties of gases
WSGG, equation (3-32). However, in the SLW model, the blackbody weights bj are expressed in terms of an absorption-line blackbody distribution function derived from the actual high- resolution spectrum. Moreover, in the SLW method the weights depend on both local temperature and local composition. The average temperature and composition are taken as reference values and therefore are important parameters in this model. The absorption coefficients aj are related to logarithmically spaced absorption cross-sections (typically 20) which are selected to span the whole range found in practical problems.
3.4.3. Grey gas EWB modelling In principle, there are three approaches to calculate total radiation properties in the context of grey gas EWB modelling. The Block Calculation Procedure (BCP)3, the Block Approximation (BA)12 method and the Band Energy Approximation (BEA)20. They are briefly described below.
3.4.3.a. Block Calculation Procedure
The Block Calculation Procedure (BCP) has been proposed by Edwards3. As a first step the band
* absorption Ak = Akωk of each band is determined. Then a constant transmittance τk is assigned to each band according to equation (3-19). However, an upper limit of 0.9 is considered as it was explained in paragraph 3.3.2.a. The width of the band is calculated from equation (3-20). In case of a symmetric band where the centre νc,k is specified, the upper and the lower limits are:
Δν Δν ν =ν + k and ν =ν − k u,k c,k 2 l,k c,k 2 (3-33)
All bands considered by the EWBM are symmetric except for the rotational band H2O which has a lower limit and the 4,3μm band of CO2 which has an upper limit (see Table 3-1). For these asymmetric bands the remaining band limit is fixed by:
Ak ν u,k −ν l,k = = Δν k (3-34) ()1−τ k
Next, the bands are blocked out spectrally and each distinct limit is numbered consecutively in ascending order of wavenumber, after assigning the index j=1 to wavenumber zero. In spectral regions where one or more bands overlap, the band transmittance is calculated as the product value of the transmittances of the overlapping bands:
34 Models for estimation of radiative properties of gases
n τ j = ∏τ k, j (3-35) k =1
The overlapping bands may be from the same absorber or from different absorbing species in the gas mixture. Where no bands absorb, the transmittance is assigned a value of unity. The total local absorptance α is now calculated from:
N α = ∑[]()()()1−τ j ()f T,ν j − f T,ν j+1 (3-36) j=1
The f function in equation (3-36) is the fraction of the black body emissive power. Analytical expressions for the calculation of the f function can be found in standard radiation textbooks (e.g. Siegel and Howell2021). Finally, the total absorption coefficient can be calculated via equation (3-4).
3.4.3.b. Block Approximation method
The second method that can be used for calculation of the total absorptance is the Block Approximation (BA) method that was first discussed by Edwards and Balakrishnan12. In this method, the total absorptance is given by:
N α = ∑[]()f ()()T,ν l, j − f T,ν u, j (3-37) j=1
As in the BCP, the temperature is set equal to the local gas temperature when local properties are calculated. The upper and lower limits in equation (3-37) are also calculated as in BCP from equation (3-33) and (3-34), but the basic difference here is that Δν k =ν u −ν l = Ak namely, the upper and lower limits of the “effective” bandwidth Ak are used rather than the “true” band limits as in BCP. In event where two or more bands are overlapping the minimum and maximum limits of the overlapping bands are used in equation (3-37), whereas the intermediate limits are disregarded.
3.4.3.c. Band Energy Approximation
The last method that is presented for the calculation of the total absorptance is the Band Energy Approximation (BEA) method20. In this method evaluation of the computationally expensive black body function is not needed. The basic assumption of the model is that the black body
35 Models for estimation of radiative properties of gases intensity is constant over each absorption band. This is an acceptable assumption when the absorption band is limited to a rather narrow spectral region. Then the total absorptance can be calculated from:
∑ Ak I b,centre π ∑ Ak I b,centre α = k = k − Δε ∞ σT 4 (3-38) I dλ ∫ b 0
Δε in the equation above is a correction factor for the overlapping bands that can be calculated as proposed in Hottel and Sarofim22.
3.5. Conclusion In practical combustion systems limiting the computational time is one of the most important issues. These systems require modelling of simultaneously occurring flow, chemical reactions and heat transfer, which make them very time consuming to simulate. Since grey gas modelling requires only one solution of the Radiative Transfer Equation (see Chapter 2) over the entire wavelength spectrum using an average absorption coefficient, it is by far the most widely used approach. The obvious drawback of the grey model is the ill correspondence between the model and the physics of radiation. In this work a comparison will be made between grey and non-grey gas calculations based on the EWBM for an industrial-scale steam cracking furnace segment (see Chapter 4). The EWBM has been chosen for this comparison since it is one of the most used and known models. Recently another model, the so-called Correlated-k (CK) model that was originally used in atmospheric physics, has gained some popularity in the research field of high temperature combustion. A first difference between the two aforementioned models is that the CK model calculates the absorption coefficient directly as compared to the EWBM of Edwards where first an expression for the effective bandwidth and the transmissivity is derived. Secondly, the CK model makes use of the idea that the absorption coefficient, which considerably varies even within small wavelength intervals, can be reordered in frequency so that it forms a smooth and monotonuously increasing function. This, in turn, makes spectral integration very straightforward as opposed to the four-region expression provided by the EWBM.
36 Models for estimation of radiative properties of gases
Overall, it can be said that the CK model has been developed more recently than the EWBM and provides some theoretical improvements. To the best of our knowledge though, no straight comparison between the two models has been conducted in the literature to prove the superiority of the CK method.
References
1 Hartmann JM, Di Leon RL, Taine J. Line-by-line and narrow-band statistical model calculations for H2O. Journal of Quantitative Spectroscopy and Radiative Transfer. 1984;30(2):119-127
2 Rothman LS, Gamache RR, Tipping RH, Rinsland, CP, Smith MAH, Benner DC, Devi VM, Flaud JM, Camy- Peyret C, Perrin A, Goldman A, Massie ST, Brown LR. The HITRAN molecular database: edition of 1991 and 1992. Journal of Quantitative Spectroscopy and Radiative Transfer. 1992;48:469-507.
3 Edwards DW. Molecular gas band radiation. Advances in Heat Transfer. 1976;12/15-93
4 Coelho P.J. Numerical simulation of radiative heat transfer from non-grey gases in three-dimensional enclosures.
5 Domoto GA. Frequency integration for radiative transfer problems involving homogeneous non-gray gases: The inverse transformation function. Journal of Quantitative Spectroscopy and Radiative Transfer. 1974;14:935-942
6 Goody R, West R, Chen L, and Crisp D. The correlated-k method for radiation calculations in non-homogeneous atmospheres. Journal of Quantitative Spectroscopy and Radiative Transfer. 1989;42:539-550
7 Soufiani A, Taine J. High temperature gas radiative property parameters of statistical narrow-band model for H2O,
CO2 and CO, and correlated-k model for H2O and CO2. International Journal of Heat and Mass Transfer. 1982;40(4):987-991
8 Elsasser WM. Heat transfer by infrared radiation in the atmosphere. Cambridge MA: Harvard University Press, 1943.
9 Goody RM. Atmospheric radiation. London and New York: Oxford University Press, 1964
10 Malkmus W. Random Lorenz band model with exponential-tailed S-1 line-intensity distribution function. Journal of the Optical Society of America. 1967;57(3):323-329
11 Soufiani, A., Hartmann, J-M, and Taine, J. Validity of Band-Model Calculations for CO2 and H2O applied tot Radiative Properties and Conductive-Radiative Transfer, Journal of Quantitative Spectroscopy and Radiative Transfer. 1985;3:243-257
12 Edwards DK, Balakrishnan A. Thermal radiation by combustion gases. International Journal of Heat and Mass Transfer. 1973;16:25-40
13 Ströhle J, Coelho PJ. On the application of the exponential wide band model to the calculation of radiative heat transfer in one- and two- dimensional enclosures. International Journal of Heat and Mass Transfer. 2002;45:2129- 2139
37 Models for estimation of radiative properties of gases
14 Cumber PS, Fairweather M, Ledin HS. Application of wide band radiation models to non-homogeneous combustion systems. International Journal of Heat and Mass Transfer. 1998;41(11):1573-1584
15 Coelho PJ, Ströhle J. Different approaches of exponential wide band model to radiative heat transfer in non- homogeneous media. Proceedings of the Second International Conference on Computational Heat and Mass Transfer. Rio De Janeiro, Brasil, October 22-26, 2001
16 Rivière P, Soufiani T, Taine J. Correlated-k fictitious gas model for H2O infrared radiation in the voight regime. Journal of Quantitative Spectroscopy and Radiative Transfer. 1995; 53(3);335-346
17 Soufiani A, Taine J. High temperature gas radiative property parameters of statistical narrow-band model for H2O,
CO2 and CO, and correlated-k model for H2O and CO2. International Journal of Heat and Mass Transfer 1997;40:987-91
18 Smith TF, Shen ZF, Friedman JN. Evaluation of coefficients for the weighted sum of grey gases model. Journal of Heat Transfer. 1982;104:602-608.
19 Denison MK, Webb BW. A spectral line-based weighted-sum-of-grey-gases model for arbitrary RTE solvers. Journal of Heat Transfer. 1993; 115:1004-1012
20 Lallemant N, Weber R. A computationally efficient procedure for calculating gas radiative properties using the exponential wide band model. International Journal of Heat and Mass Transfer. 1996;39:3273-3286
21 Siegel R, Howell JR. Thermal radiation heat transfer. New York: McGraw-Hill, Inc., 1972
22 Hottel HC, Sarofim AF. Radiative transfer. New York: McGraw-Hill, Inc., 1967.
38 Chapter 4
Comparison of grey gas and non-grey gas radiation models
4.1. Introduction In this chapter, simulation results of grey and non-grey gas, CFD, simulations of an industrial naphtha cracking furnace are presented. As discussed before, because of computational load most radiation models applied in CFD simulations of practical applications are based on the grey gas simplification, as opposed to the actual non-grey “real” gas. It is the main aim of this chapter to investigate if this simplification is justifiable. This will be done by quantifying the differences in results of grey and non-grey simulations concerning important predicted variable profiles, like the flue gas flow and temperature profile, as well as the heat fluxes to the tubes. To simulate the thermal radiation exchange, both in grey and non-grey calculations, the Finite Volume (FV) scheme, implemented in the FLUENT software package, has been used (see Chapter 2). Both the grey and non-grey radiation models are based on the Exponential Wide Band Model (EWBM) of Edwards. Since FLUENT provides the possibility to do simulations using both grey and non-grey models, but doesn’t support the EWB model, two separate User Defined Functions (UDF’s) have been programmed to incorporate the EWBM into the FLUENT simulations. Although both models have been thoroughly discussed in Chapter 3, this chapter contains an additional paragraph on the practical implementation of both models. First, the geometry and operating conditions of the steam cracking furnace segment are discussed.
39 Comparison of grey gas and non-grey gas radiation models
Convection section
Outlet
C Wall
Burners
A Wall
Reactor tubes (a) (b)
Tube 3
Tube 1 Tube 2 (c)
Figure 4-1: Furnace segment geometry. (a) Front view, (b) 3D-view, (c) top view.
4.2. Furnace segment geometry and operating conditions Since the main goal of this chapter is a comparative study between grey and non-grey radiation models only a representative segment of an industrial naphtha cracking furnace with radiation burners will be simulated. A schematic representation of this segment is given in Figure 4-1. By considering two sides as symmetry planes, the furnace segment represents an infinitely long furnace. The position of the three reactor tubes in this segment is chosen in such a way that the distances between the tubes and the symmetry planes are representative for the complete furnace.
40 Comparison of grey gas and non-grey gas radiation models
14 radiation burners are located at the walls of the furnace segment. One row of burners is located in the front wall (A wall) and another row of burners is located at the rear wall (C wall) of the furnace. The convection section outlet is located at the top of the C wall and runs across its entire length. Due to a high degree of turbulent mixing between fuel and oxidizer in the burner cups, complete combustion of the fuel gas in the cups themselves is reasonably assumed. This entails that only hot combustion products (hot flue gas) enters the radiation section of the furnace. The product species concentration remains invariant in the simulated domain. 96578 cells are used to discretize the physical space between the segment walls, the symmetry planes and the reactor tubes. No coupled furnace reactor simulations are performed as only a furnace segment is simulated. Therefore, a fixed internal tube skin temperature is applied and is considered to be part of the boundary conditions of the simulations. The basic simulation conditions, the furnace segment dimensions and an overview of the boundary conditions and material properties are given in Table 4-1.
Table 4-1: Simulaton conditions of the steam cracking furnace segment.
Geometry Boundary conditions Radiation Section Fuel gas flow rate (kg/s) 1.23242 Height (m) 7.32 Fuel gas composition (wt %)
Lenght (m) 0.7 CO2 0.122
Width (m) 1.7 H2O 0.118
Thickness of refractory (m) 0.23 O2 0.0366
Thickness of insulation (m) 0.05 N2 0.7234 Thickness of casing (m) 0.005 Fuel gas inlet temperature (K) 1900 Number of burners 14 Fuel gas inlet pressure (atm) 1 Reactor Fuel gas pressure drop (Pa) 30 Number of reactor tubes 3 Internal tube skin temperature (K) 1100 External tube diameter (m) 0.1319 Material properties Internal tube diameter (m) 0.1143 Emissivity of furnace wall (-) 0.85 Convection Section Emissivity of tube skin (-) 0.6 Height (m) 2.514 Thermal conductivity of refractory wall (W/mK) 0.394 Lenght (m) 0.7 Thermal conductivity of insulation (W/mK) 0.19 Width (m) 1.126 Thermal conductivity of casing (W/mK) 56 Outlet Radiation Section Thermal conductivity of tube skin (W/mK) 26.05 Height (m) 0.534 Depth (m) 0.453
41 Comparison of grey gas and non-grey gas radiation models
4.3. Practical implementation of the radiation models In both the grey and non-grey model the four-region expression has been used to calculate the band transmittance (equation 3-19) and bandwidth (equation 3-20), while taking into account the upper limit of 0.9 for the calculation of the band transmittance as suggested by Edwards1.
4.3.1. Grey gas modelling As mentioned in Chapter 3 there are three approaches to calculate total radiation properties while applying grey gas EWB modelling: the Block Calculation Procedure (BCP), the Block Approximation (BA) method and the Band Energy Approximation (BEA). Since little difference in results can be detected when all three methods are compared the computationally less demanding Band Energy Approximation has been used for grey gas modelling in this work.
Nine bands (five bands for CO2 and four bands for H2O) are taken into account when calculating the overall absorption coefficient. These bands are:
CO2: 15μm, 10.4μm, 9.4μm, 4.3μm, 2.7μm
H2O: 6.3μm, 1.87μm, 1.38μm, 2.7μm Since both gases have overlapping absorbing bands at 2.7μm the following relation, proposed by Hottel and Sarofim2, has been applied to calculated the band absorptance of the overlapping band of width Δη :
A A A = A + A − a b a+b a b Δη (4-1)
The width Δη is calculated from the approximate band limits for a parallel-plate geometry2.
The mean beam length L=0.047m that is used in equation (3-4) to calculate the local absorption coefficient at a control volume is set equal to the average characteristic cell size in the domain, which is determined by the total volume and the total number of cells in the CFD simulations. Although this choice has the disadvantage that the final outcome is somewhat dependent on the grid resolution, it is consistent with the EWBM variant which has been developed for isothermal gas radiation. This condition is fulfilled in a cell volume. It is noted that the temperature field in a steam cracking furnace is highly non-uniform due to the significant heat sink in the middle of the furnace where the tubes are suspended.
42 Comparison of grey gas and non-grey gas radiation models
1,6
1,4 L=0,05 m 1,2 9 grey bands (CO2 + H2O) 5 grey bands (CO2 + H2O) 1
0,8
0,6
Absorption coefficient[1/m] 0,4 L=1,0 m
0,2 L=3,0 m
0 1000 1200 1400 1600 1800 2000 Temperature [K]
Figure 4-2: Total absorption coefficients as calculated by the EWBM for three different mean beam lengths considering 9 and 5 absorbing/emitting bands.
4.3.2. Non-grey gas modelling Since the FLUENT software package only supports the assumption of four or less distinct absorption bands, a selection of four out of nine bands of CO2 and H2O, that will finally be considered, has been made. It was found, that the set of bands given below yield a satisfactory approximation of the absorption coefficient for the conditions met in a steam cracking furnace over a wide range of mean beam lengths (see Figure 4-2). These bands are:
CO2: 15μm, 4.3μm, 2.7μm
H2O: 6.3μm, 2.7μm For the overlapping 2.7μm band again the relation proposed by Hottel and Sarofim2 was used. A second limitation of non grey gas modelling in FLUENT is the inability of the program to dynamically change the bandwidth limits during calculations. Therefore an average width for each band is computed in a preprocessing step. These values are then given to FLUENT as fixed bandwidth limits. In this preprocessing step only temperature dependency of the bandwidth is taken into account. This is acceptable since in practice the flue gas composition in the furnace remains unaltered – complete combustion in the burner caps is assumed – and the pressure drop
43 Comparison of grey gas and non-grey gas radiation models
Table 4-2: Division of the wavelength spectrum into 4 absorbing bands and 5 clear windows.
Lower limit Upper limit Band Type [μm] [μm] 1 0 2.55 Clear Window 2 2.55 2.78 Overlap 2.7 μm 3 2.78 4.15 Clear Window 4 4.15 4.47 CO2 4.3 μm 5 4.47 5.54 Clear Window 6 5.54 7.17 H2O 6.3 μm 7 7.17 12.77 Clear Window 8 12.77 18.14 CO2 15 μm 9 18.14 45 Clear Window
600 CO2 15μm 550 H2O 6,3μm CO2 4,3μm 500 Overlap 2,7μm
450
400
350
300 bandwidth [cm-1] bandwidth
250
200
150
100 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 Temp [K]
Figure 4-3: The bandwidth as calculated by the EWBM for the four absorbing/emitting bands. (L=0.047m) in a typical steamcracking furnace does not exceed 50 Pa (Oprins and Heyndrickx3). Figure 4-3 shows the bandwidth for the four absorbing and emitting bands that are considered, over the temperature range of 1100 to 1900 K. The values shown in this graph have been used to calculate an average bandwidth representative for each band. It is noted that the abrupt change in trend that can be noticed for the H2O 6.3μm and CO2 15μm curve is due to a shift from the linear region (low temperatures) of the four region expression to the square root region (high temperatures). For the spectral “clear” windows, outside the limits of the absorbing bands, the absorption coefficient is set equal to zero. To make sure that the entire wavelength spectrum is covered by
44 Comparison of grey gas and non-grey gas radiation models
4 the bands, the FLUENT manual proposes the following limits: λmin = 0 and λmaxTmin = 50000 .
Here λmin and λmax are the minimum and maximum wavelength bounds of the wavelength spectrum and Tmin is the minimum expected temperature in the domain, which is taken equal to 1100K in this work. The computed band limits that are used in the context of non-grey modelling are quoted in Table 4-2.
4.4. CFD modelling approach A three-dimensional mathematical model is used to simulate the flow in the steam cracking furnace segment. The model consists of the partial differential equations describing the conservation of momentum, heat and mass, in combination with two equations for turbulence modelling. If the dependent variable is denoted by Φ, the general differential equation is:
∂ ()ρΦ + div (ρvrΦ − Γ gradΦ )= S ∂t Φ Φ (4-2) where ΓΦ is the diffusion coefficient and SΦ is the source term. The turbulence kinetic energy and the dissipation of turbulence kinetic energy are calculated with the k-ε model5. To solve these partial differential equations along with the boundary and inlet conditions, the finite volume method using a segregated solver is used. This segregated solver is used in combination with the SIMPLE algorithm for pressure-velocity coupling and the first order upwind differencing discretization scheme for the convection operator. The above approach together with the DO-FV radiation model are embodied in the general FLUENT CFD program4. The models for calculating the total and band local absorption coefficients for the grey and non-grey gas simulations respectively are implemented in User Defined Functions (UDF’s) and plugged into the FLUENT solver. Opaque, grey diffuse furnace wall and tube skin surfaces are assumed in both grey and non-grey gas calculations. The index of refraction is set to 1 and scattering is neglected. Finally, two theta (polar angle) and two phi (azimuthal angle) divisions are found to be adequate for the problem studied.
45 Comparison of grey gas and non-grey gas radiation models
4.5. Results and discussion As mentioned in the introduction, the goal of this work is to quantify the effect of the grey gas simplification on important furnace design parameters like the temperature distribution in the fire box, the flue gas flow pattern and the heat fluxes directed from the flue gas to the tubes. These parameters determine the process of naphtha cracking in the reactor coils and other detrimental side effects as the coke formation on the tube’s internal surface.
(a) (b)
z=2.995 z=2.995
Figure 4-4: Temperature contour plots in a vertical cross section in the middle of the furnace at y=0.35m. (a) grey gas radiation model. (b) non-grey gas radiation model. First, in Figure 4-4, the temperature contour plot for the grey and non-grey gas calculations in a vertical plane in the middle of the furnace at y=0.35m (see Figure 4-1) and a horizontal plane at a furnace height of z=2.995m (fifth burner) are shown. One can observe that the flue gas temperature calculated with the non-grey gas model is higher than that calculated with the grey gas model. A more clear description of the temperature distribution in the firebox is given in Figure 4-5, where the horizontal temperature profile across the x-direction at y=0.35m at a height of z=3.775m is presented. The temperature line connects the centre of the 6th burner of the front
46 Comparison of grey gas and non-grey gas radiation models
1900
1800
1700
1600
1500 Temperature [K] Temperature 1400
1300
1200 grey gas model z = 3.775 m non-grey gas model 1100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 X [m]
Figure 4-5: Temperature profile along the width of the furnace at a height z=3.775m.
(A) wall with the centre of the corresponding burner on the rear (C) wall. Both modelling approaches predict decreasing flue gas temperatures when leaving the burners’ area and approaching the centre (with respect to the x-direction) of the furnace. In the centre of the furnace, where the reactor tubes are situated, a steep temperature drop, the so-called “heat sink”, is predicted in both cases. However, the use of a non-grey gas radiation model results in a different flue gas temperature profile between the front wall and the rear wall. It is observed in Figure 4-5 that the temperature lines start diverging from one another while moving away from the wall and converge again in the “heat sink” zone close to the tubes. As in Figure 4-4, it is also remarked that higher gas temperature values are calculated with the non-grey gas model along the entire x-axis. The calculated temperature differences, which exceed 150K in some areas, are more pronounced in the zones lying between x=0.3m and x=0.8m, approximately, to the left of the tube and between x=1.05m and x=1.4m, approximately, to the right of the tube (outlet side). Higher temperatures in the furnace domain, when using the non-grey gas model, imply that less radiation is emitted by the flue gas, possibly because the non-grey gas is limited to radiation in distinct bands opposed to the grey gas which can radiate energy over the complete wavelength spectrum. This argument is supported by Figure 4-6, which is presented in the same fashion as Figure 4-5. In Figure 4-6 horizontal radiation temperature profiles at y=0.35m and at the same
47 Comparison of grey gas and non-grey gas radiation models
1750
1700
1650
1600
1550
1500 Radiation temperature [K] temperature Radiation
1450
1400 grey gas model z = 3.775 m non-grey gas model 1350 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 X [m]
Figure 4-6: Radiation temperature profiles along the width of the furnace at a height of z=3.775m.
height as the static temperature profiles of Figure 4-5 are shown. Radiation temperature of a body is the temperature that a blackbody of similar dimensions would have if it radiates the same intensity at the same frequency and is defined by:
1 4 ⎛ G ⎞ θ R = ⎜ ⎟ (4-3) ⎝ 4σ ⎠ where G is the incident radiation, namely the total radiation energy that arrives at a location per unit time per unit area and is defined by:
G = IdΩ ∫ (4-4) Ω=4π
Lower radiation temperatures, when using the non-grey gas model all along the x-axis (Figure 4-6), indicate lower values for the incident radiation, being the numerator in equation (4-3), all along the x-axis too. The later entails that less radiation energy, G, is emitted by the gas resulting in higher temperatures in the furnace domain than those that are calculated with the grey model (see Figure 4-4 and Figure 4-5). On the contrary, when using the grey gas model more radiation is emitted by the flue gas, and thus, more radiation is transferred to the reactor tubes. This is
48 Comparison of grey gas and non-grey gas radiation models
200 Tube 1 180
160
140
120
100
80 Heat Flux to[kW/m2] tube Flux Heat 60
40
20 grey gas model non-grey gas model 0 012345678 X [m]
Figure 4-7: Tube heat flux profiles along height (z-direction). Solid lines: grey gas model. Dashed lines: non-grey gas model.
shown in Figure 4-7, where the heat flux profiles to tube 1 (see Figure 4-1) with the grey gas and the non-grey gas model are plotted. The line representing the heat flux profile with the non-grey gas model is lower than that representing the heat flux profile with the grey gas model along the tube height with the exception of a short zone lying between 6m height and the top of the tube. In both the grey and non-grey case the heights of heat flux peaks correspond with the heights of the burners inside the furnace walls (Figure 4-7). The descending heat flux profile trend versus tube height that is predicted with both models is due to the fact that the distance between successive burners increases with height and also because the highest burner is situated at 4.549m height whereas the total furnace height is 7.32m. Overall, the predicted furnace thermal efficiency namely, the fraction of the heat input in the furnace that goes to the tubes, rises from 37.6% when using the non-grey gas model to 43.9% when using the grey gas model. This difference of 6.3% in thermal efficiency that is predicted is quite large considering the scale and the importance of the industrial process and should be taken into account by the furnace designer. It should be mentioned here that the exact thermal efficiency values will be somewhat different when a complete furnace is simulated, coupled with a reactor model for the steam cracking
49 Comparison of grey gas and non-grey gas radiation models
(a) (b)
z=2.995 z=2.995
Figure 4-8: Velocity contour plots in a vertical cross section in the middle of the furnace at y=0.35m. (a) grey gas radiation model. (b) non-grey gas radiation model. process. However, the qualitative and quantitative assessment of the grey gas approximation for steam cracking furnace calculations can be carried out effectively in the context of furnace segment simulations. Full industrial-scale coupled furnace-reactor simulations using non-grey gas radiation modelling will be presented in the Chapter 5. This paragraph is completed with a discussion on the flow patterns in the furnace as they are predicted when using the grey and non-grey gas radiation models. This discussion is supported by Figure 4-8 and Figure 4-9. Figure 4-8 shows the contour plots of velocity magnitude in a vertical plane in the middle of the furnace at y=0.35m in the same way as the temperature contours in Figure 4-4. Figure 4-9 shows horizontal profiles of velocity magnitude at a height of z=3.775m. Although Figure 4-8 shows little difference between the two models, quantitative differences in the predicted velocity profiles do exist and are shown in Figure 4-9. An increase in velocity magnitude can be noticed when the non-grey model is compared to the grey model. This
50 Comparison of grey gas and non-grey gas radiation models
6
5
4
3 Velocity [m/s]
2
1
grey gas model z = 3.775 m non-grey gas model 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 X [m]
Figure 4-9: Flue gas velocity profiles along the width of the furnace at a height of z=3.775m.
increase becomes more pronounced when the tube zone is approached. A possible explanation for this increase is the higher gas temperature in case of the non-grey gas modelling. A higher gas temperature results in a lower density of the gas, and thus, higher gas velocities are required to achieve the same overall mass flow.
4.6. Conclusion The target of this Chapter was to quantify the effect of the grey gas approximation that is commonly used for simulations of industrial applications on important variable profiles like the flue gas flow and temperature as well as the reactor tube heat flux. These are important parameters for the optimal design and operation of the furnace. The comparison between the two models shows that when the grey gas simplification is used more energy is emitted by the flue gas in the furnace box, and thus, more energy is transferred to the process gas in the reactor tubes. Thus, the predicted thermal efficiency increases from 37.6% when using the non-grey gas model to 43.9% when using the grey gas model. This 6.3% difference in the predicted thermal efficiency is quite large considering the scale and the importance of the industrial process and
51 Comparison of grey gas and non-grey gas radiation models should be taken into account by the furnace designer. It has also been shown that, although both models reproduce identical basic characteristics of the flow pattern in the furnace, noticeable quantitative differences in the flue gas speed are predicted in some regions of the furnace domain.
References
1 Edwards DW. Molecular gas band radiation. Advances in Heat Transfer. 1976;12/15-93
2 Hottel HC, Sarofim AF. Radiative Heat transfer. New York: McGraw-Hill, Inc., 1967
3 Oprins AJM, Heynderickx GJ. Calculation of three-dimensional flow and pressure fields in cracking furnaces. Chemical Engineering Science. 2003;58:4883-4893
4 Fluent 6.2 Users’s Guide. Fluent Inc., Lebanon, NH. January 2005.
5 Launder BE, Spalding DB. Lectures in mathematical models of turbulence. London: Academic Press. 1972
52 Chapter 5
Investigation of high-emissivity coatings in steam cracking furnaces
This chapter is structured according to a typical format of a research paper. Some overlap between this chapter and the previous ones unavoidably exists.
5.1. Abstract The efficiency of the application of high-emissivity coatings on the furnace walls in steam cracking technology can only be evaluated on the basis of a description of radiative heat transfer in frequency bands. To this end, a non-grey gas radiation model based on the Exponential Wide Band Model (EWBM) has been developed and applied in the context of three-dimensional CFD simulations of an industrial naphtha cracking furnace with side-wall radiation burners. The simulation results show that applying high-emissivity coatings on the furnace walls improves the thermal efficiency of the furnace, the naphtha conversion and the ethylene yield.
5.2. Introduction Steam cracking of hydrocarbons to olefins is an endothermal process carried out in tubular reactor coils suspended in large gas fired furnaces. Due to the high temperature of the combustion gases, radiation is the predominant mode of heat transfer in the furnace. Considering the scale and the importance of the industrial process even a small rise in thermal efficiency (fraction of the furnace heat input that is absorbed by the tubes) can be translated into an important increase in the olefin yields or an important decrease in the required fuel gas input. Applying high-
53 Investigation of high-emissivity coatings in steam cracking furnaces emissivity coatings on the furnace walls is believed to result in an increase in thermal efficiency1. This is based on the fact that radiation in gases is fundamentally different than radiation from surfaces. Surfaces absorb and emit radiation at all frequencies. Gases absorb and emit radiation at certain discrete frequencies. Due to various overlapping effects, these absorption lines form absorption bands. Spectral intervals in which no radiation is absorbed or emitted, the so-called clear windows, are situated in between the absorption bands. The application of a high-emissivity coating on a furnace wall first implies that the amount of radiation energy that is reflected by the wall decreases. Next, the amount of radiation energy that is absorbed by the wall increases. Since the furnace walls are insulated and the heat loss to the environment is small, more energy is re- radiated back in the furnace. In contrast to the reflected energy that preserves its spectral character, the re-radiated energy is spread over the entire wavelength spectrum. This entails that when the emissivity of the material, by which the wall is coated, increases, an additional amount of radiation energy originating from the flue gas bands will be absorbed by the wall and re- radiated back in the furnace partly through clear windows. This additional amount of radiation that travels via the clear windows can reach the reactor tubes without being influenced by the gas. The target in this chapter is to investigate the aforementioned concept by quantifying the relevant radiation fluxes on the furnace walls, on the reactor tubes and in the flue gas. For that reason, two coupled furnace/reactor simulations are performed with different furnace wall emissivities. The furnace calculations are based on the Computational Fluid Dynamic (CFD) approach using commercial software (FLUENT)2. In order to assess the high-emissivity coating concept, a non- grey (banded) gas radiation model is needed. To that end, a variant of the Exponential Wide Band Model (EWBM) of Edwards3 is programmed as a stand-alone code and is plugged into the CFD solver as a User-Defined Function (UDF). The reactor calculations are based on a plug flow reactor model in combination with a detailed reaction network for steam cracking of hydrocarbons. Details on the modelling and numerical procedures as well as the furnace geometry and operating conditions are given in the next paragraphs.
5.3. Exponential Wide-Band Model (EWBM)
5.3.1. Calculation of band transmittance The EWBM3 provides a mathematical model to correlate experimental data and to predict wide band properties. It is based on the assumption that the rotation lines in the band are equally
54 Investigation of high-emissivity coatings in steam cracking furnaces spaced and can be reordered in frequency so that they form an array with exponentially decreasing line intensities starting from the band centre. Three parameters are required to specify the radiative properties:
∞ The integrated band intensity: α = (S / d)d(ν −ν ) ∫ 0 0
The exponential decay width: ω and the mean line width to spacing parameter β = πγ 0 / d
The mean line intensity to spectral line spacing ratio is given by equation (5-1) in case of an asymmetric band with the upper limit ν u specified, by equation (5-2) in case of an asymmetric band with the lower limit ν l specified or by equation (5-3) in case of a symmetric band with the band centre ν c specified.
−(ν −ν l ) / ω S / d = (α /ω)e (5-1)
−(ν u −ν ) / ω S / d = (α /ω)e (5-2)
−2 (ν −ν ) / ω S / d = (α /ω)e c (5-3)
4 -2 -1 -3/2 -1/2 It is noted that in early papers concerning the EWBM, C1 (cm atm ), C2 (cm atm ) and C3
-1 -1 -1 (cm ) are used instead of α (m kg ), γ (-) and ω (m ). C1 corresponds to α , C2 is related to γ
2 via the relationship γ = C2 /(4C1C3 ) and C3 corresponds to ω .
Theoretically,α , β and ω vary with temperature and pressure within each band. The EWBM model however assumes that they vary with respect to temperature only. The pressure dependence is taken into account through the Pe number (equation (5-4)).
n Pe = {}[]P / P0 [1+ (b −1)x] (5-4)
Expressions for the functional dependence of α , β and ω have been developed elsewhere3,5 and are summarized below.
m 1− exp − ± u δ Ψ(T) [ ( ∑k=1 k k )] α(T ) = α 0 (T ) m 1− exp − ± u δ Ψ(T ) (5-5) []()∑k=1 0,k k 0
55 Investigation of high-emissivity coatings in steam cracking furnaces
−1/ 2 β = β 0 (T /T0 ) Φ(T ) / Φ(T0 ) (5-6)
m ∞ (υ + g + δ −1)! ∏ ∑ k k k e −ukυk υ =υ (g −1)!υ ! Ψ(T ) = k =1 k 0,k k k m ∞ (υ + g −1)! (5-7) ∏ ∑ k k e −ukυk k =1 υk =0 (g k −1)!υ k !
1/ 2 2 ⎧ m ∞ ⎫ ⎪ ⎡(υk + g k + δ k −1)! −u υ ⎤ ⎪ ⎨ ⎢ e k k ⎥ ⎬ ∏ ∑ (g −1)!υ ! ⎪ k=1 υk =υ0,k ⎣ k k ⎦ ⎪ Φ(T ) = ⎩ ⎭ (5-8) m ∞ (υ + g + δ −1)! ∏ ∑ k k k e −ukυk k=1 υk =υ0,k (g k −1)!υk !
1/ 2 ω = ω0 (T /T0 ) (5-9)
where uk = c2ν k /T and u0,k = c2ν k /T0 (5-10)
The subscript 0 denotes some reference condition. T0 = 100 K is considered for convenience. υ0,k denotes the lowest possible initial state and is taken equal to zero when δ k is positive or zero, and equal to the absolute value of δ k when δ k is negative. The set of quantum numbers
(υ1 ,υ2 ,...,υk ,...,υ m ) describes the molecular, vibrational state before absorption of a photon of radiation energy.
After absorption, a set of quantum numbers (υ1 + δ1 ,υ2 + δ 2 ,...,υk + δ k ,...,υm + δ m ) describes the th th transition between different modes, creating the j absorption band for the i species. The gk values are statistical weights for generate and degenerate vibrations. The ν 1 ,ν 2 ,...,ν k ,...,ν m values are wavenumbers associated with the characteristic molecular vibrations. Values for the exponential wide band model parameters participating in equations (5-5) to (5-10) 3 can be found in Table III of Edwards for six species (H2O, CO2, CO, NO, SO2 and CH4) and all their absorption bands.
5.3.1.a. Four region expression
Originally, Edwards3 provided correlations to a body of experimental data for the calculation of the band transmittance. The set of correlations consists of relations of linear, square root and
56 Investigation of high-emissivity coatings in steam cracking furnaces logarithmic form with respect to the optical depth at the band head depending on the level of the absorption strength of the band. The so-called four region expression equations (5-11) to (5-14) are given below.
* A = τ H for τ H ≤ 1, τ H ≤ η (5-11)
* 1/ 2 A = (4ητ H ) −η for η ≤ τ H ≤ 1/η , η ≤ 1 (5-12)
* 1/ 2 A = ln(τ Hη) + 2 −η for 1/η ≤ τ H ≤ ∞ , η ≤ 1 (5-13)
* A = lnτ H +1 for τ H ≥ 1, η ≤ 1 (5-14)
* where A = A/ω is the dimensionless band absorption, τ H = αX /ω is the maximum optical depth at the band head and η = βPe . The band absorption or "effective" bandwidth A can be interpreted as the width of a black band (i.e. completely absorbing band) centered about the middle of the real absorption band and absorbing the same amount of radiative energy as the real band does. The band transmittance is calculated from
* τ H dA τ k = * (5-15) A dτ H and the bandwidth from
Δv = A/(1−τ k ) (5-16) Use of equation (5-15) implies a grey gas assumption for each band, an assumption that breaks down at small path lengths. For that reason Edwards3 suggests an upper limit for the calculation of the band transmittance (equation (5-17)):
τ = min(τ ,0.9) k k (5-17)
Imposing an upper limit for the band transmittance may introduce serious mistakes if a recursive relationship is used due to the strong dependence on the grid resolution6. One way to mitigate the problem is to define the band limits from the average properties of the domain in a preprocessing step. Then the band transmittance can be calculated from equation (5-18):
τ = 1− A/ Δν k fix (5-18)
Using the latter approach, the band absorption A is still calculated from the four-region expression (equations (5-11) to (5-14)) and the temperature and composition dependence are still
57 Investigation of high-emissivity coatings in steam cracking furnaces taken into account. However, instead of calculating the band transmittance of equation (5-15) in a first step, the bandwidth Δν fix is computed from the average properties of the domain in a first step.
5.3.1.b. Integration method
Cumber et al.7 suggest that in order to avoid the grey gas assumption, which is necessary in order to calculate the band transmittance through equation (5-15), one could calculate the spectral transmittance, employing an analytical expression according to the exponential-tailed band model as:
−(S / d )i X j ∑ 1+(S / d ) ( X /η ) τ = e ij i j (5-19) v , X = ρL
where i denotes the band and j the participating species. (S / d)i values are calculated from relations (5-1) to (5-3). Summation over i is necessary when two or more bands of the same absorbing component overlap. Summation over j is necessary when two or more bands of different absorbing components overlap. Next, the average band transmittance is computed via integration of the spectral transmittance over the band wavelength spectrum:
1 τ = τ dν k Δν ∫ ν (5-20) Δν
The selection of the band limits in equation (5-20) does not significantly influence the results8. In turn, the spectral absorption coefficient is calculated from:
(S / d) ρ κ = i j ν ∑ (5-21) ij 1+ (S / d)i (X /η) j because it is linked to the spectral transmittance by:
−κν L τν = e (5-22)
For very small path lengths, (S / d)(X /η) << 1, equation (5-20) and equation (5-22) can be simplified into equation (5-23) and equation (5-24) respectively:
58 Investigation of high-emissivity coatings in steam cracking furnaces
∑ −(S / d )i ρ j ij τ v = e (5-23)
κν = ∑(S / d)i ρ j ij (5-24)
The last relation shows that for very small path lengths, grid independent solutions can be obtained by using this approach. On the other hand, a problem arising when employing this method is that the EWBM parameters given by Edwards3 muster the best agreement with experimental data when they are used in conjunction with the four-region expression. When these parameters are used for numerical quadrature of equation (5-20), discrepancies in the band absorption values, calculated with the four-region expression method and with the integration method, are observed. The problem can be alleviated by adjusting ω upward by 20% when spectral integration is performed, as suggested by Edwards and Balakrishnan5 and by Edwards3. It is noted that adjustment of the ω value should be applied outside the linear region namely, τ H ≤ η ≤ 1.
5.3.2. Non-grey gas modelling Four absorption bands are considered in this work. In these bands carbon dioxide and water vapor are regarded as the only components of the combustion gases that absorb and emit radiation. These bands are:
CO2: 15μm , 4.3μm , 2.7μm
H2O: 6.3μm , 2.7μm
Distinct absorption coefficients are derived for the distinct absorption bands. In brief, the band transmittance (equation (5-15)) and the band width (equation (5-16)) are first calculated over the range 1000-1900 K using the four region expression. Next, an average width for each band is computed and remains fixed for the recalculation of the band transmittances over the range 1000-1900 K using equation (5-18). It is noted here that the computed bandwidths do not change considerably with temperature and thus a simple arithmetic average bandwidth is considered representative for each band. The reason for applying this approach is to bypass the imposition of an upper arbitrary limit for the band transmittance as explained in paragraph 5.3.1. The fixed bandwidth is also used for the computation of fixed band
59 Investigation of high-emissivity coatings in steam cracking furnaces
Table 5-1: Division of the wavelength spectrum into 4 gas absorption bands and 5 gas clear windows.
limits over which the RTE is solved. In case of a symmetric band where the band centre ν c,k is specified, the upper and the lower limits are
ν =ν + Δν / 2 ν =ν − Δν / 2 u,k c,k k and l,k c,k k (5-25)
All bands considered by the EWBM are symmetric except the rotational band of H2O that has a lower limit and the 4,3μm band of CO2 that has an upper limit. For these asymmetric bands the remaining band limit is calculated from equation (5-26).
v −ν = A /(1−τ ) = Δν u,k l,k k k k (5-26) Finally, the Beer's law (see equation (5-22)) is applied to calculate the band absorption coefficient via equation (5-27):
1 κ = − ln()τ L k (5-27) For the spectral "clear" windows, other than the absorption bands, the absorption coefficient is set equal to zero. To make sure that the entire wavelength spectrum is covered by the bands we 2 choose λmin = 0μmK and λmaxTmin = 50000μmK . Hereλmin and λmax are the minimum and the maximum wavelength bounds of the wavelength spectrum and Tmin is the minimum expected temperature in the domain, which is taken equal to 1000 K in this work. The computed band limits are quoted in Table 5-1. The mean beam length L , needed for the calculation of the absorption coefficient, is calculated based on the average cell volume in the calculation domain. An alternative is the use of a mean
60 Investigation of high-emissivity coatings in steam cracking furnaces beam length for the whole enclosure9. The first choice has the disadvantage that the final outcome is somewhat dependent on the grid resolution but is consistent with the EWBM variant, which is applied here and has been developed for isothermal gas radiation. The latter condition is only fulfilled in a cell volume and can not be assumed for the entire furnace domain.
5.4. Overview of the furnace/reactor calculations
5.4.1. Reactor Model In the plug flow reactor model, a set of mass balances for the process gas species is solved simultaneously with the energy and the pressure drop equation. The steady-state mass balance for a component j in the process gas mixture over an infinitesimal volume element with cross sectional surface area Ω , circumference ω , and length dz is
dF ⎛ nR ⎞ j = ⎜ n r ⎟Ω dz ⎜∑ kj k ⎟ (5-28) ⎝ k=1 ⎠ The energy equation is given by
dT ∑∑Fj c pj = ωq + Ω rk (−ΔH k ) (5-29) jkdz
The pressure drop equation, accounting for friction and changes in momentum, is given by
dp ⎛ 2 f ζ ⎞ du ⎜ ⎟ 2 = α⎜ + ⎟ρ g u + αρ g u (5-30) dz ⎝ d t πrb ⎠ dz
The process gas temperature profile, conversion, and concentration profiles can be calculated based on an imposed heat-flux profile or external tube skin temperature profile. In this article, calculations are performed using a heat flux profile obtained from a CFD furnace calculation. The reactor model that is described above is coupled to a detailed reaction network for the steam cracking of hydrocarbons, containing over 1000 reactions among 128 species. A detailed description of the reactor model and the reaction mechanism for the steam cracking of hydrocarbons can be found in Van Geem et al.10 and Heynderickx and Froment11.
61 Investigation of high-emissivity coatings in steam cracking furnaces
5.4.2. Furnace Model
5.4.2.a. Flow
The calculation of the steady-state flue gas flow pattern and temperature profile in the furnace is based on the Reynolds-averaged Navier Stokes equations. The continuity, momentum and energy conservation equations are
∇ ⋅ ()ρυr = 0 (5-31)
∇ ⋅ ()ρυrυr = −∇p + ∇ ⋅ (τ )+ ρgr (5-32)
⎛ r ⎞ ∇ ⋅ (υr(ρE + p)) = ∇ ⋅⎜k ∇T − h J + τ ⋅υr ⎟ + q ⎜ eff ∑ j j ()eff ⎟ rad (5-33) ⎝ j ⎠ The Discrete Ordinates (DO) radiation model, solved with a Finite-Volume (FV) procedure, is used to predict the radiative heat flux contribution to the energy equation. The standard k − ε model12 is used for the calculation of the turbulent properties:
⎛⎛ μ ⎞ ⎞ ∇ ⋅ ρυrk = ∇ ⋅⎜⎜ μ + t ⎟∇k ⎟ + G + G − ρε () ⎜⎜ ⎟ ⎟ k b (5-34) ⎝⎝ σ k ⎠ ⎠
⎛⎛ μ ⎞ ⎞ ε ε 2 ∇ ⋅ ρυrε = ∇ ⋅⎜⎜ μ + t ⎟∇ε ⎟ + C G + C G − C ρ () ⎜⎜ ⎟ ⎟ 1ε ()k 3ε b 2ε (5-35) ⎝⎝ σ ε ⎠ ⎠ k k
Due to high degree of turbulent mixing between fuel and oxidizer in the side-wall burner cups, complete combustion of the fuel gas in the cups themselves is a reasonable assumption. This entails that only hot combustion products (hot flue gas) enter the radiation section of the furnace that is simulated in this work. The product species concentrations remain invariant in the simulated furnace domain. Therefore, no explicit combustion calculations are to be performed in the context of the overall CFD modelling approach.
5.4.2.b. Radiation
To simulate the thermal radiation exchange in the non-grey gas calculations, a conservative variant of the Discrete Ordinates (DO) radiation model, called the Finite-Volume (FV) scheme and implemented in the FLUENT software package, is used. The finite volume method was originally introduced by Raithby and Chui13 and is slightly modified and described in detail by
62 Investigation of high-emissivity coatings in steam cracking furnaces
Baek et al.14 and Raithby.15 In this method, the inflow and outflow of radiant energy across control volume faces are balanced with attenuation and augmentation of radiant energy within a control volume and a control angle. The total solid angle, 4π steradians, is discretized into a finite number of discrete solid (control) angles in any convenient manner, depending on the problem being dealt with. The number of equations to be solved for each control volume is therefore equal to the number of control angles specified. The main advantage of using the finite volume method is that it allows radiation to be treated in a way similar to the methods used in the CFD calculations i.e. the concept of strict conservation of radiant energy, the way boundary conditions are applied, and the formation and solution of the discrete equations, are common for all processes: fluid flow, convective heat transfer, and radiation. The DO model considers the RTE in the direction sr as a field equation:
σT 4 σ 4π ∇ ⋅ (I(r, sr)sr) + (κ + σ )I(r, sr) = κn 2 + s I(r, sr′)Φ (sr ⋅ sr′)dΩ ′ s ∫ (5-36) π 4π 0
When modelling non-grey gas radiation the equation above is solved band-wise. For the spectral r r intensity I λ (r, s) the following equation is obtained:
σ 4π ∇ ⋅ (I (r, sr)sr) + (κ + σ )I (r, sr) = κ n 2 I + s I (r, sr′)Φ (sr ⋅ sr′)dΩ ′ λ λ s λ λ bλ ∫ λ (5-37) 4π 0
Here κ λ is the spectral absorption coefficient, and I bλ is the black body intensity given by the Planck function. The scattering coefficient, the scattering phase function, and the refractive index n are assumed to be independent of the wavelength. The non-grey gas implementation divides the radiation spectrum into N wavelength intervals. The RTE is integrated over each wavelength interval, resulting in transport equations for the quantity I λ Δλ . The behaviour in each absorption band is assumed to be grey. The black body emission in the wavelength band per unit solid angle is:
4 2 σT I bλ = ( f (nλuT ) − f (nλlT))n (5-38) π where λu and λl are the upper and the lower band wavelength limits respectively and f (nλT ) is the fractional black body function. Finally, the total intensity in each direction sr and position r is computed using equation (5-39):
63 Investigation of high-emissivity coatings in steam cracking furnaces
I(r, sr) = I (r, sr)Δλ ∑ λκ κ κ (5-39)
The summation is performed over all the wavelength bands.
• Radiation boundary conditions: In case of grey gas radiation modelling the radiative flux leaving a surface is:
2 4 qout = (1− ε w )qin + n ε wσTw (5-40) where ε w is the grey wall emissivity, qin is the incident radiative heat flux at the surface that is calculated as:
q = I sr ⋅ nrdΩ in ∫ in (5-41) sr⋅nr>0
The boundary conditions in case of non-grey/DO modelling are applied on a band basis. The treatment of the boundary conditions within a grey band is the same for grey gas radiation modelling. The radiative flux leaving a surface is:
2 4 qout,λ = (1− ε wλ )qin,λ + ε wλ ( f (nλuT ) − f (nλlT ))n σTw (5-42) where ε w,λ is the wall emissivity in the band and qin,λ is the incident radiative heat flux at the surface, within the band wavelength interval Δλ , which is calculated as:
q = Δλ I sr ⋅ nrdΩ in,λ ∫ in,λ (5-43) sr⋅nr>0
5.4.3. Coupled furnace/reactor simulations In order to evaluate the effect of high wall emissivity coatings on the furnace thermal efficiency, coupled furnace (fire-side) - reactor (process-side) calculations are necessary. A schematic diagram of the complete simulation procedure is shown in Figure 1. In the CFD furnace calculations the set of partial differential equations, along with the boundary and inlet conditions, is solved with the finite control volume method using a segregated solver and the SIMPLE algorithm for pressure-velocity coupling. As mentioned above the DO-FV method is used for non-grey gas radiation modelling. The local band absorption coefficients are calculated by means of the EWBM that has been implemented in a User Defined Function (UDF). The latter is plugged into the CFD solver. The sequence of steps in the CFD furnace calculations (see Figure 5-1) is repeated several times until a converged solution is found. The CFD furnace simulation
64 Investigation of high-emissivity coatings in steam cracking furnaces provides the heat flux profile along the reactor length, which in turn forms the input for the reactor simulator. Next, the reactor simulation updates the process gas temperature profile and the heat transfer coefficient profile on the process gas side. Those two profiles are set as boundary conditions for the CFD furnace calculations. This two-way coupling is repeated several times until convergence of both the furnace and the reactor simulation is obtained.
Figure 5-1: Overview of the coupled furnace/reactor calculations.
5.5. Furnace geometry and operating conditions The main design specifications and operating conditions of the simulated industrial naphtha cracking furnace are summarized in Table 5-3 and Table 5-2 respectively. The side and the top views of the simulated furnace radiation section are given in Figures 2 and 3 respectively. Due to the symmetry consideration only half of the actual furnace is simulated. Eight reactor coils with seven passes each, are suspended in two staggered rows in the furnace. The composition of the
65 Investigation of high-emissivity coatings in steam cracking furnaces
Table 5-2: Furnace and reactor operating conditions.
Table 5-3: Furnace specifications.
66 Investigation of high-emissivity coatings in steam cracking furnaces
Figure 5-2: Front view of the industrial steam cracking furnace.
Figure 5-3: Top view of the industrial steam cracking furnace.
67 Investigation of high-emissivity coatings in steam cracking furnaces
Table 5-4: Solid wall emissivity values for 17 wavelength bands (Jackson and Yen, 1994)
naphtha that is cracked in these coils is summarized in Table 5-3. For the reactor calculations, a feed of 9 different paraffins, 31 isoparafines, 18 naphthenes and 8 aromatics is taken into account. The furnace is heated by means of 224 radiation burners positioned in 16 rows of burners in the front wall (A wall) and 16 rows of burners in the rear wall (C wall). The furnace outlet is located at the top of the C wall and runs across its entire length. Due to a high degree of turbulent mixing between fuel and oxidizer in the burner cups, complete combustion of the fuel gas in the cups themselves is a reasonable assumption. This entails that only hot combustion products (hot flue gas) enter the radiation section of the furnace. The flue gas species concentration (Table 5-3) remains invariant in the simulated domain. 653,836 cells are used to discretize the furnace domain between the segment walls, the symmetry plane and the reactor tubes.
5.6. Results and discussion The goal of this work is to investigate the influence of the furnace wall emission coefficient on the heat fluxes to the reactor tubes and on the thermal efficiency of the entire furnace. The fluxes determine the process gas temperature inside the tubes and thus the naphtha conversion and olefin yields. An improvement in thermal efficiency of the furnace can be translated into an increase of the yields or a decrease of the required fuel input.
68 Investigation of high-emissivity coatings in steam cracking furnaces
Two coupled furnace/reactor simulations are performed. The two simulations differ only with respect to the wall emissivity value. Opaque, grey-diffuse furnace wall and tube skin surfaces are assumed in both cases. The index of refraction is set to 1 and scattering is neglected. In the first simulation (case 1): the furnace wall emissivityε w = 0. 386 . In the second simulation (case 2) the wall is coated and an increased furnace wall emissivity value is applied in the calculations:
ε w = 0.738 . The grey wall emissivities are calculated from the non-grey values according to:
17 ∑ε w,z ⋅ sum z = ε w,grey (5-44) z=1 17 ∑ sum z = 1 (5-45) z=1 The emissivity values of the solid wall material for 17 different wavelength bands, considered in this work, are given in Table 5-416,17. The fraction of the black body emissive power for the band z is given by sum z.
60000 18% Case 1: εw=0.386 Case 2: εw=0.738 50000 % difference between Case 1 and Case 2
13%
] 40000 2
30000 8%
20000 % Difference
10000 3%
0 123456789 -2% -10000 Band
Average net suface incident radiation flux [W/m flux radiation incident suface net Average -20000 -7%
-30000
-40000 -12%
Figure 5-4: Average net surface incident radiation on the coils of Reactor 1 (Figure 2) in each band for cases 1 and 2 and % difference in the net surface incident radiation fluxes between the two cases. Bands n° 1, 3, 5, 7 and 9 are clear windows. Bands n° 2, 4, 6 and 8 are absorption bands.
69 Investigation of high-emissivity coatings in steam cracking furnaces
The simulations show an average increase in the total heat flux to the reactor tubes of 2.1% when the wall emissivity changes from 0.386 to 0.738. This results in an overall rise in thermal efficiency from 40.0% to 40.9%. In order to fully understand this difference, the surface in- and outgoing radiation flux per band for both the furnace wall and the reactor tubes need to be considered. First, in Figure 5-4, a comparison of the average net surface incoming radiation flux to reactor 1 for each spectral band is shown. The net surface incoming radiation flux is defined by:
2 4 qnet,in,λ = qin,λ − qout,λ = ε wλ qin,λ −ε wλ ( f (nλuT) − f (nλl T ))n σTw (5-46)
In Figure 5-4 the numbers on the horizontal axis correspond to the band numbers as they are given in Table 5-1. It is reminded that bands n° 1, 3, 5, 7 and 9 are clear windows, which means that the flue gas in these bands is non-absorbing and non-emitting. On the other hand, bands n° 2, 4, 6 and 8 are absorption bands in which the flue gas does absorb and emit radiation. It can readily be seen that there is a net positive radiation flux to the tubes in every band, and, when comparing the simulation results between case 1: ε w = 0.386 and case 2: ε w = 0.738 , a different trend in the radiative fluxes is observed. Increasing the wall emission coefficient from 0.386 to 0.738 makes the net fluxes towards the reactor tubes rise in the clear windows and drop in the absorption bands. Addition of these two reverse effects gives a total increase of 2198 W/m² or 2.28% in the net incoming radiation flux. It is this rise in the net surface incoming radiation to the reactor tubes that results in an increase of the furnace thermal efficiency. Since no changes are made to the boundary conditions of the reactor tubes or the models that are used for the simulation, the explanation of the observations described above should be sought on the furnace wall side of the fire box. Figure 5-5 shows the average net surface outgoing radiation flux from the furnace wall for each band. This is defined in equation (5-47):
2 4 qnet,out,λ = qout,λ − qin,λ = ε wλ ( f (nλuT ) − f (nλlT ))n σTw − ε wλ qin,λ (5-47)
One can notice that in the clear windows there is a positive net outgoing radiation flux whereas in the absorption bands there is a negative net outgoing radiation flux that is equivalent to a positive net incoming radiation flux. Furthermore, the change in the furnace wall emission coefficient from ε w = 0.386 to ε w = 0.738 decreases the net outgoing radiation from the wall through the
70 Investigation of high-emissivity coatings in steam cracking furnaces
8000 Case 1: εw=0.386 60% Case 2: εw=0.738 % difference between Case 1 and Case 2 6000 40% ] 2
4000
20% 2000 % Difference
0 0% 123456789*
Band -2000 -20%
-4000 Average net surface outgoing radiation flux [W/m flux radiation outgoing surface net Average -40% -6000
* For visual convenience the net radiation difference for band 9 (+148,4%) is not shown
-8000 -60%
Figure 5-5: Average net surface outgoing radiation from the furnace wall in each band for cases 1 and 2 and % difference in the net surface outgoing radiation fluxes between the two cases. Bands n° 1, 3, 5, 7 and 9 are clear windows. Bands n° 2, 4, 6 and 8 are absorption bands.
90000 Black body emission Incident radiation 80000
70000
60000 ] 2 50000
40000 Heat flux[W/m
30000
20000
10000
0 123456789 Band
Figure 5-6: Comparison between the black body emission flux at the average wall temperature (1378 K) and the average incident radiation flux on the wall surface in each band. Bands n° 1, 3, 5, 7 and 9 are clear windows. Bands n° 2, 4, 6 and 8 are absorption bands.
71 Investigation of high-emissivity coatings in steam cracking furnaces absorption bands and increases the net outgoing radiation from the wall through the clear windows. This explains why when increasing the furnace wall emissivity the reactor tubes receive more radiation energy through the clear windows and less radiation energy through the absorption bands. An explanation for the furnace wall effects is presented in Figure 5-6. This figure shows the blackbody emission at the average wall temperature (1378 K) and the average surface incident radiation for all bands in case 1. It is observed that in the clear windows, the blackbody emission is always higher than the surface incident radiation. This is expected because the average surface incident radiation in clear windows is only influenced by the furnace wall itself, the reactor tubes – at lower temperature – and the heat loss at the furnace outlet through the clear windows. More specifically, since part of the radiation energy emanating from a wall surface through the clear windows is absorbed by the tubes and is lost at the furnace outlet (no absorption/emission takes place in the clear windows), it is physically impossible that the average surface incident radiation is higher than the maximum possible amount of emitted energy, which is the blackbody emission, at the average wall temperature. On the other hand, in the absorption bands, the reverse phenomenon occurs. The average blackbody emission is lower than the average furnace surface incident radiation. This is a consequence of the presence of the absorbing and emitting gas. The flue gas being at higher temperature than the furnace wall emits additional radiation energy towards the furnace wall, which results in a higher average furnace surface incident radiation flux as compared to the blackbody emission at the average furnace wall temperature. Finally, it becomes evident by inspecting equation (5-47) that, taking into account that the average band
2 4 blackbody emission flux ( ( f (nλuT ) − f (nλlT ))n σTw ) is higher than the average band incident radiation flux ( qin,λ ) in the clear windows and lower than it in the absorption bands, the net outgoing radiation will be positive in the clear windows and negative in the absorbing bands. Furthermore an increase of the furnace wall emissivity value changes the two terms on the right handside in equation (5-47) and results in an increase in the net outgoing radiation from the furnace wall in the clear windows and a decrease in the net outgoing radiation from the furnace wall in the absorption bands. Both effects were shown in Figure 5-5. So far, the effect of the furnace wall emissivity coating on the in- and outgoing band radiation fluxes at the furnace wall and tube surfaces have been discussed in Figure 5-4 to Figure 5-6.
72 Investigation of high-emissivity coatings in steam cracking furnaces
700000
650000 ] 2 600000
550000 Incident radiation [W/m
500000
Case 1: εw=0.386
Case 2: εw=0.738 450000 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 Reactor width [m]
Figure 5-7: Horizontal profile of the sum of incident radiation fluxes in the clear windows along the width of the furnace between the A and C walls (see Figure 3). Height=3.775 m and length=1.732 m.
350000
330000
310000
290000 Incident radiation [W/m²] 270000
250000
Case 1: εw=0.386 Case 2: εw=0.738 230000 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 1,6 1,7 Furnace width [m]
Figure 5-8: Horizontal profile of the sum of incident radiation fluxes in the absorption bands along the width of the furnace between the A and C walls (see Figure 3). Height=3.775 m and length=1.732 m.
73 Investigation of high-emissivity coatings in steam cracking furnaces
Figure 5-7 and Figure 5-8 provide the "link" between the "furnace wall-effect" and the "tube- effect". In Figure 5-7, typical horizontal profiles of the sum of incident radiation energies in the clear windows along the furnace width for the cases 1 and 2 are presented. The profiles are taken at a height of 3.775 m and a length of 1.732 m. They start from the furnace A wall in-between two burners and end on the opposite wall. The corresponding profiles for the sum of incident radiation energies in absorption bands are presented in Figure 5-8. It can be seen that a rise in the furnace wall emission coefficient makes the incident radiation rise in the clear windows (Figure 5-7) and fall in the absorption bands (Figure 5-8) all along the path from the walls to the tubes This corresponds with the results shown in Figure 5-4: a rise in the net surface incoming radiation to the reactor tubes through the clear windows and a fall in the net surface incoming radiation to the reactor tubes through the absorption bands when increasing the wall emissivity. Furthermore, the profiles in Figure 5-7 and Figure 5-8 are in consistency with the profiles presented in Figure 5-5. More specifically, the rise in incident radiation travelling through the clear windows when increasing the wall emissivity (Figure 5-7) corresponds with the rise in the net outgoing radiation leaving the furnace wall through the clear windows when increasing the wall emissivity (Figure 5-5). On the other hand, the fall in incident radiation travelling through the absorption bands when increasing the wall emissivity (Figure 5-8) corresponds with the drop in the net outgoing radiation from the furnace wall through the absorption bands when increasing the wall emissivity (Figure 5-5). Finally, it is noted that the gradual decrease in the incident radiation fluxes, as the centre of the furnace is approached (Figure 5-7 and Figure 5-8), is due to the "heat sink" in the zones where the reactor tubes are suspended. Overall, it is concluded that the physical mechanism that determines this rise in thermal efficiency is the reallocation of radiation energy in clear windows and absorption bands on the furnace wall. Irradiative energy originating from the flue gas (radiated from the absorption bands) is partially converted, due to wall emission, into outgoing radiation in the clear windows. A higher furnace wall emission coefficient will enhance this energy reallocation effect on the furnace walls by increasing the outgoing (emitted) radiation through the clear windows and decreasing the total outgoing (reflected+emitted) radiation through the absorption bands. Since radiation travelling through the clear windows can reach the reactor tubes without partially being absorbed by the flue gas the overall heat flux towards the reactors rises.
74 Investigation of high-emissivity coatings in steam cracking furnaces
100 1150
90 naphtha Conversion ethylene yield 1100 80 propylene yield methane yield 70 1050
process gas temperature [K] Temperature 60 1000 50 950 40
30 900 Conversion and yield [wt%] 20 850 10
0 800 0 102030405060 Reactor length [m]
Figure 5-9: Typical reactor simulation results for the most important variable profiles along the length of
reactor 4 (see Figure 2). ε w = 0.738 , reactor operating conditions: Table 5-2.
Table 5-5: Simulation results
The rise in furnace thermal efficiency when increasing the wall emissivity has an influence on the naphtha conversion and the product yields. Figure 5-9 shows typical process gas temperature profiles, naphtha conversion and product yield profiles along the tube length of reactor 4. A comparison of the most important simulation results between case 1 and case 2 for one of the reactors in the furnace is presented in Table 5-5. The furnace thermal efficiency rises from
75 Investigation of high-emissivity coatings in steam cracking furnaces
40.0% to 40.9% by applying a high emissivity coating on the furnace wall (case 2). As a result, the naphtha conversion rises from 93.4 wt% to 94.5 wt% and the ethylene yield rises from 24.1 wt% to 24.6 wt%. Similar results are calculated for reactors the other reactors (not shown). These differences are small, but considering the industrial importance and scale of the thermal cracking process, significant.
5.7. Conclusions The efficiency of the application of high-emissivity coatings on the furnace walls in steam cracking technology can only be evaluated on the basis of a description of radiative heat transfer in frequency bands. To this end, a non-grey gas radiation model based on the Exponential Wide Band Model (EWBM) is developed and applied in the context of three-dimensional CFD simulations of an industrial naphtha cracking furnace with side-wall radiation burners. The simulation results show that applying high emissivity coatings on the furnace walls improves the thermal efficiency of the furnace, and the cracking results. The increase in thermal efficiency should be attributed to the energy reallocation mechanism in clear windows and absorption bands taking place on the furnace walls. Wall surface incident radiation originating from absorption bands is partially converted due to wall emission into wall surface outgoing radiation through the clear windows. Applying a high-emissivity coating on the furnace wall increases the net outgoing radiation at the wall through the clear windows and decreases the net outgoing radiation from the wall through the absorption bands. Since radiation travelling through clear windows can reach the reactor tubes without partially being absorbed by the flue gas, contrary to radiation travelling through absorption bands, the thermal efficiency of the furnace increases.
References
1 Hellander JC. Throughput enhancement and tube temperature stabilization.Hydrocarbon Processing. 1997;76(10):91-96.
2 Fluent 6.2 User's Guide. Fluent Inc., Lebanon, NH. January 2005.
3 Edwards DW. Molecular gas band radiation. Advances in Heat Transfer. 1976;12:15-93.
4 Hottel HC, Sarofim AF. Radiative Heat Transfer. New York: McGraw-Hill, Inc., 1967
5 Edwards DK, Balakrishnan A. Thermal radiation by combustion gases. International Journal of Heat and Mass Transfer. 1973;16:25-40.
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6 Ströhle J, Coelho PJ. On the application of the exponential wide band model to the calculation of radiative heat transfer in one- and two-dimensional enclosures. International Journal of Heat and Mass Transfer. 2002;45:2129- 2139.
7 Cumber PS, Fairweather M, Ledin HS. Application of wide band radiation models to non-homogeneous combustion systems. International Journal of Heat and Mass Transfer. 1998;41(11):1573-1584.
8 Coelho PJ, Ströhle J. Different approaches of the exponential wide band model to radiative heat transfer in non- homogeneous media. Proceedings of the Second International Conference on Computational Heat and Mass Transfer. Rio De Janerio, Brasil, October 22-26, 2001.
9 Liu F, Becker HA, Bindar Y. A comparative study of radiative heat transfer modelling in gas-fire furnaces using simple grey gas and weighted-sum-of-grey-gases models. International Journal of Heat and Mass Transfer. 1998;41:3357-3371.
10 Van Geem KM, Heynderickx GJ, Marin GB. Effect of radial temperature profiles on yields in steam cracking. AIChE Journal. 2004;50(1):173-183.
11 Heynderickx GJ, Froment GF. A pyrolysis furnace with reactor tubes of elliptical cross section. Industrial Engineering & Chemistry Research. 1996;35:2183-2189.
12 Launder BE, Spalding DB. Lectures in mathematical models of turbulence. London: Academic Press, 1972.
13 Raithby GD, Chui, EH. A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. Journal of Heat Transfer. 1990;112:415-423.
14 Baek SW, Kim MY, Kim JS. Nonorthogonal finite-volume solutions of radiative heat transfer in a three- dimensional enclosure. Numerical Heat Transfer, Part B. 1998;34:419-437.
15 Raithby GD. Discussion of the finite-volume for radiation, and its application using 3D unstructured meshes. Numerical Heat Transfer, Part B. 1999;35:389-405.
16 Heynderickx GJ, Nozawa M. Banded gas and nongray surface radiation models for high-emissivity coatings. AIChE Journal. 2005;51(10):2721-2736.
17 Jackson JD, Yen CC. Ceramics in Energy Applications Conference. London: Pergamon Press; 1994.
77 Chapter 6
General conclusion
In this work, three-dimensional CFD simulations of a steam cracking furnace with radiation burners have been performed. To perform these simulations the commercial CFD software package FLUENT is used. The focus is on modelling of radiative heat transfer since this is the most important mode of heat transport in the radiative section of a steam cracking furnace. While the variations of radiation properties for opaque solids are fairly smooth, gas properties exhibit very irregular wavelength dependencies. As a result, two general types of models have been developed to describe the radiative properties of a gas: grey and non-grey gas models. Non-grey gas models take into account the wavelength dependencies of the radiative properties of gases while grey gas models provide total radiative property values independent of the wavelength interval. A first practical part of this work focuses on a comparative study between grey and non-grey radiations models when applied in the context of an industrial scale steam cracking furnace segment. Both models are based on the Exponential Wide Band Model (EWBM) of Edwards and implemented into FLUENT by use of User Defined Functions (UDF). The target is to quantify the effect of the grey gas approximation on important predicted variable profiles like the flue gas flow and temperature as well as on the reactor tube heat flux. These are important parameters for the optimal design and operation of the furnace. By comparing the results obtained with the two models it is found that when the grey gas simplification is used more energy is emitted by the flue gas in the furnace box, and thus, more energy is transferred to the process gas in the reactor tubes. The predicted thermal efficiency increases from 37.6% when using the non-grey model to 43.9% when using the grey gas model. This 6.3% difference in the predicted thermal efficiency is quite large considering the scale and the importance of the industrial process and should be taken into account by the furnace designer. Furthermore, both models give identical basic
78 General conclusion characteristics for the flow pattern in the furnace, but noticeable quantitative differences in the flue gas speed are predicted in some regions of the furnace domain. A second part of this work is to perform a coupled full furnace-reactor simulation. This is done by linking FLUENT with the in-house reactor simulation program COILSIM. These coupled simulations are used to investigate the influence of the furnace wall emissivity on the heat fluxes to the reactor tubes and on the thermal efficiency of the entire furnace. The fluxes significantly determine the cracking process inside the coils (e.g. naphtha conversion and olefin yields) and an improvement in thermal efficiency can be translated into an increase of the product yields or a decrease in the required fuel input. To assess the impact of the furnace wall emission coefficient, two coupled furnace-reactor simulations are performed using a non-grey gas radiation model (EWBM of Edwards) with two different grey wall emissivities. In the first grey wall emissivity simulation the emission coefficient is set to a value of 0.386 and in the second simulation the wall emissivity is set to 0.738. Both values are derived from non-grey wall emissivity values. The simulation results show that applying high emissivity coatings on a furnace wall improves the thermal efficiency of the furnace, and the cracking results. The increase in thermal efficiency is attributed to the energy reallocation mechanism in the clear windows and absorption bands taking place at the furnace walls. Wall surface incident radiation originating from absorption bands is partially converted due to furnace wall emission into wall surface outgoing radiation through the clear bands. It is found that applying a high-emissivity coating on the furnace wall increases the net outgoing radiation at the wall through the clear bands and decreases the net outgoing radiation from the wall through the absorption bands. Since radiation travelling through clear windows can reach the reactor tubes without partially being absorbed by the flue gas, contrary to radiation travelling through absorption bands, the thermal efficiency of the furnace increases.
79