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Eulerian-Lagrangian method

5 Mathematical model

5.2 Eulerian-Lagrangian method

Figure 5.1: Solid volume fraction fluctuation with time. Dimensionless bed height

= 0.5, dimensionless bed width = 0.75, dimensionless gas velocity = 2.

5.2 Eulerian-Lagrangian method

CPFD (Computational Particle Fluid Dynamic) model is one of the latest developments using Eulerian-Lagrangian method. The method blends discrete Lagrangian and continuum Eulerian method [68]. The CPFD method solves fluid and particle conservation equations in three dimensions treating the fluid field as Eulerian and the particles as Lagrangian. There is a strong coupling between the fluid and the particles.

Thermal and chemistry calculations are available for the fluid and particle phases coupled for energy and reaction purposes. The CPFD is incorporated with Multiphase-Particle in cell (MP-PIC). In the MP-PIC method, conservation equations are solved for the continuous phase. For solid phase a transport equation is solved for the particle distribution function [16, 69]. The short description of gas and particle equations are given in this chapter referring the literature references [56, 70, 71] and more details are found in th.

Gas phase mass conservation is given by Equation 5.26 [70]:

πœ•πœ•οΏ½π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”οΏ½

πœ•πœ•π‘π‘ +βˆ‡ βˆ™ οΏ½π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”π‘’π‘’οΏ½βƒ—π‘”π‘”οΏ½=π›Ώπ›Ώπ‘šπ‘šΜ‡π‘π‘ (5.26)

where 𝛼𝛼𝑔𝑔 is gas volume fraction (void fraction), πœŒπœŒπ‘”π‘” is gas density, 𝑒𝑒�⃗𝑔𝑔 is the gas velocity, π›Ώπ›Ώπ‘šπ‘šΜ‡π‘π‘ is the gas mass production rate per volume from the particle-gas chemistry . The momentum conservation equation for the gas phase is given as:

πœ•πœ•οΏ½π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”π‘’π‘’οΏ½βƒ—π‘”π‘”οΏ½

πœ•πœ•π‘π‘ +βˆ‡ βˆ™ οΏ½π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”π‘’π‘’οΏ½βƒ—π‘”π‘”π‘’π‘’οΏ½βƒ—π‘”π‘”οΏ½

=βˆ’π›Όπ›Όπ‘”π‘”βˆ‡π‘π‘π‘”π‘”+𝐹𝐹⃗+π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”π‘”π‘”βƒ—+βˆ‡ βˆ™(π›Όπ›Όπ‘”π‘”πœπœπ‘”π‘”)

(5.27)

46 CHAPTER 5. MATHEMATICAL MODEL

where 𝑝𝑝𝑔𝑔 is gas pressure, 𝑔𝑔⃗ is acceleration due to gravity, 𝐹𝐹⃗ is the rate of interphase momentum transfer per unit volume and πœπœπ‘”π‘”is gas stress tensor. The constitutive equation for the gas stress is given in index notation as:

πœπœπ‘”π‘”,π‘˜π‘˜π‘–π‘– =πœ‡πœ‡ οΏ½πœ•πœ•π‘’π‘’π‘˜π‘˜

πœ•πœ•π‘šπ‘šπ‘–π‘– +πœ•πœ•π‘’π‘’π‘–π‘–

πœ•πœ•π‘šπ‘šπ‘˜π‘˜οΏ½ βˆ’2

3πœ‡πœ‡π›Ώπ›Ώπ‘˜π‘˜π‘–π‘–πœ•πœ•π‘’π‘’π‘˜π‘˜

πœ•πœ•π‘šπ‘šπ‘˜π‘˜ (5.28)

where πœ‡πœ‡ is shear viscosity. The shear viscosity is the sum of laminar shear viscosity and turbulence viscosity based on the Smagorinsky turbulence model. In the model, large eddies are directly calculated. The unresolved sub grid turbulence is modeled by using eddy viscosity. The turbulence viscosity is given as:

πœ‡πœ‡π‘‘π‘‘ =πΆπΆπœŒπœŒπ‘”π‘”βˆ†2οΏ½οΏ½πœ•πœ•π‘’π‘’π‘˜π‘˜

πœ•πœ•π‘šπ‘šπ‘–π‘– +πœ•πœ•π‘’π‘’π‘–π‘–

πœ•πœ•π‘šπ‘šπ‘˜π‘˜οΏ½

2

(5.29)

where 𝐢𝐢 is sub grid eddy coefficient and known as Smagorinsky coefficient. In the simulation of bubbling fluidized bed gasification reactor the coefficient is used with a constant value of 0.01. The sub grid length is given by the relation, βˆ†=

(βˆ†π‘šπ‘šβˆ†π‘¦π‘¦βˆ†π‘ π‘ )1/3. The energy equation for the gas phase is given by:

πœ•πœ•οΏ½π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”β„Žπ‘”π‘”οΏ½

πœ•πœ•π‘π‘ +βˆ‡ βˆ™ οΏ½π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”β„Žπ‘”π‘”π‘’π‘’οΏ½βƒ—π‘”π‘”οΏ½

=βˆ’π›Όπ›Όπ‘”π‘”οΏ½πœ•πœ•π‘π‘

πœ•πœ•π‘π‘ +π‘’π‘’οΏ½βƒ—π‘”π‘”βˆ™ βˆ‡π‘π‘π‘”π‘”οΏ½+βˆ… βˆ’ βˆ‡ βˆ™ οΏ½π›Όπ›Όπ‘”π‘”π‘žπ‘žβƒ—οΏ½+𝑄𝑄̇

+π‘†π‘†β„Ž +π‘žπ‘žπ·π·Μ‡

(5.30)

where β„Žπ‘”π‘” is the gas enthalpy, βˆ… is viscous dissipation, 𝑄𝑄̇ is energy source per unit volume, π‘†π‘†β„Ž is conservative energy exchange from solid phase to the gas phase, π‘žπ‘žβƒ—

is gas heat flux and π‘žπ‘žπ·π·Μ‡ is enthalpy diffusion term. The gas heat flux π‘žπ‘žβƒ— is calculated as:

π‘žπ‘žβƒ— =βˆ’πœ†πœ†π‘”π‘”βˆ‡π‘‡π‘‡π‘”π‘” (5.31)

where πœ†πœ†π‘”π‘” is gas thermal conductivity. The thermal conductivity is the sum of molecular conductivity and eddy conductivity. The eddy conductivity is determined from Prandtl number as:

𝑃𝑃𝐴𝐴𝑑𝑑 =πΆπΆπ‘π‘πœ‡πœ‡π‘‘π‘‘

πœ†πœ†π‘‘π‘‘ (5.32)

The standard value of Prandtl number used in the model is 0.9.

The enthalpy diffusion term is given by:

5.2 EULERIAN – LAGRANGIAN METHOD 47

π‘žπ‘žΜ‡π·π· =οΏ½ βˆ‡οΏ½β„Žπ‘˜π‘˜π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”π·π·βˆ‡π‘Œπ‘Œπ‘”π‘”,π‘˜π‘˜οΏ½

𝑁𝑁𝑠𝑠 π‘˜π‘˜=1

(5.33)

The mixture enthalpy is related to the species enthalpy by:

β„Žπ‘”π‘” =οΏ½ π‘Œπ‘Œπ‘”π‘”,π‘˜π‘˜ 𝑁𝑁𝑠𝑠 π‘˜π‘˜=1

β„Žπ‘˜π‘˜ (5.34)

where the summation is all gas species π‘π‘π‘˜π‘˜. The species enthalpy depends on the gas temperature and expressed by:

β„Žπ‘˜π‘˜ =οΏ½ 𝐢𝐢𝑇𝑇1 𝑝𝑝,π‘˜π‘˜π‘‘π‘‘π‘‡π‘‡

𝑇𝑇0 βˆ†β„Žπ‘šπ‘š,π‘˜π‘˜ (5.35)

where βˆ†β„Žπ‘šπ‘š,π‘˜π‘˜ is the heat of formation at reference temperature 𝑇𝑇0 and 𝐢𝐢𝑝𝑝,π‘˜π‘˜ is the specific heat at constant pressure for species i. The equation of state for an ideal gas is used to determine the pressure:

𝑝𝑝=πœŒπœŒπ‘”π‘”π‘…π‘…π‘‡π‘‡π‘”π‘”οΏ½ π‘Œπ‘Œπ‘”π‘”,π‘˜π‘˜

π‘€π‘€π‘€π‘€π‘˜π‘˜

𝑁𝑁𝑠𝑠 π‘˜π‘˜

(5.36)

where R is universal gas constant and π‘€π‘€π‘€π‘€π‘˜π‘˜ is the molecular weight of the species i.

A gas can be a mixture of different species. A transport equation is solved for each of the gas species and the total fluid phase properties are calculated from the species mass fraction. The transport equation for the individual species in the gas phase is given by:

πœ•πœ•οΏ½π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”π‘Œπ‘Œπ‘”π‘”,π‘˜π‘˜οΏ½

πœ•πœ•π‘π‘ +π›π›οΏ½π·π·π›Όπ›Όπ‘”π‘”πœŒπœŒπ‘”π‘”π‘Œπ‘Œπ‘”π‘”,π‘˜π‘˜π‘’π‘’οΏ½βƒ—π‘”π‘”οΏ½

=βˆ’βˆ‡ βˆ™ οΏ½πœŒπœŒπ‘”π‘”π·π·π›Όπ›Όπ‘”π‘”βˆ‡π‘Œπ‘Œπ‘”π‘”,π‘˜π‘˜οΏ½+π›Ώπ›Ώπ‘šπ‘š Μ‡ π‘˜π‘˜,π‘π‘β„Žπ‘’π‘’π‘šπ‘š

(5.37) π‘Œπ‘Œπ‘”π‘”,π‘˜π‘˜ is the mass fraction of each gas species and π›Ώπ›Ώπ‘šπ‘š Μ‡ π‘˜π‘˜,π‘π‘β„Žπ‘’π‘’π‘šπ‘š is the net production rate of species due to gas phase chemical reactions. 𝐷𝐷 is the turbulent mass diffusion rate which is related to viscosity by Schmidt number. The default value of Schmidt number is 0.9 in this work.

πœ‡πœ‡π‘”π‘”

πœŒπœŒπ‘”π‘”π·π· =𝑆𝑆𝑝𝑝 (5.38)

48 CHAPTER 5. MATHEMATICAL MODEL

MP-PIC method calculates the particle phase dynamics using the particle distribution function (PDF), 𝑓𝑓𝑠𝑠. A transport equation is solved for the PDF. The transport equation for 𝑓𝑓𝑠𝑠 is given by [72]: drag function. The drag function depends on the particle size, velocity, position and time as shown in Equation 5.41. The particle size is expressed as particle radius instead of diameter. Wen-Yu drag model is used in the CPFD model. Although, the Syamlal & O’Brien model is used in the CFD model in this work, there was no possibility of using the same drag model in CPFD simulation due to the restrictions in the CPFD solver Barracuda VR 14.1. The Syamlal & O’Brien model is not included in Barracuda and the solver does not allow the users to define the model.

𝐷𝐷𝑠𝑠 =𝐢𝐢𝐷𝐷3

The particle movement equation is:

π‘‘π‘‘π‘šπ‘šβƒ—π‘ π‘ 

𝑑𝑑𝑝𝑝 =𝑒𝑒�⃗𝑠𝑠 (5.44)

5.2 EULERIAN – LAGRANGIAN METHOD 49 The particle volume fraction is defined by 𝑓𝑓𝑠𝑠 is:

𝛼𝛼𝑠𝑠 =οΏ½ π‘“π‘“π‘ π‘ π‘šπ‘šπ‘ π‘ 

πœŒπœŒπ‘ π‘  π‘‘π‘‘π‘šπ‘šπ‘ π‘ π‘‘π‘‘π‘’π‘’οΏ½βƒ—π‘ π‘ π‘‘π‘‘π‘‡π‘‡π‘ π‘  (5.45)

The sum of volume fraction of the gas and solid phase is unity:

𝛼𝛼𝑔𝑔+𝛼𝛼𝑠𝑠= 1.0 (5.46)

The interphase momentum transfer included in the Equation 5.27 is:

𝐹𝐹⃗ =οΏ½ π‘“π‘“π‘ π‘ οΏ½π‘šπ‘šπ‘ π‘ οΏ½π·π·π‘ π‘ οΏ½π‘’π‘’οΏ½βƒ—π‘”π‘” βˆ’ 𝑒𝑒�⃗𝑠𝑠� βˆ’βˆ‡p

πœŒπœŒπ‘ π‘ οΏ½+π‘’π‘’οΏ½βƒ—π‘ π‘ π‘‘π‘‘π‘šπ‘šπ‘ π‘ 

𝑑𝑑𝑝𝑝 οΏ½ π‘‘π‘‘π‘šπ‘šπ‘ π‘ π‘‘π‘‘π‘’π‘’οΏ½βƒ—π‘ π‘ π‘‘π‘‘π‘‡π‘‡π‘ π‘  (5.47) It is assumed that no heat is released inside the particles during the chemical reaction. This means that the temperature is constant inside the particles when they undergo chemical reaction. Moreover, it is assumed that the heat released at the particle surface does not affect the surface energy balance significantly.

The relation for the particle to gas phase conservative energy exchange equation is:

π‘†π‘†β„Ž =οΏ½ π‘“π‘“π‘ π‘ οΏ½π‘šπ‘šπ‘ π‘ οΏ½π·π·π‘ π‘ οΏ½π‘’π‘’οΏ½βƒ—π‘”π‘”βˆ’ 𝑒𝑒�⃗𝑠𝑠�2βˆ’ 𝐢𝐢𝑣𝑣𝑑𝑑𝑇𝑇𝑠𝑠 𝑑𝑑𝑝𝑝�

βˆ’π‘‘π‘‘π‘šπ‘šπ‘ π‘ 

𝑑𝑑𝑝𝑝 οΏ½β„Žπ‘ π‘ +1

2οΏ½π‘’π‘’οΏ½βƒ—π‘ π‘ βˆ’ 𝑒𝑒�⃗𝑔𝑔�2οΏ½οΏ½ π‘‘π‘‘π‘šπ‘šπ‘ π‘ π‘‘π‘‘π‘’π‘’οΏ½βƒ—π‘ π‘ π‘‘π‘‘π‘‡π‘‡π‘ π‘ 

(5.48)

where β„Žπ‘ π‘  is particle enthalpy. The lumped heat equation for the particle is:

𝐢𝐢𝑣𝑣𝑑𝑑𝑇𝑇𝑠𝑠 𝑑𝑑𝑝𝑝 = 1

π‘šπ‘šπ‘ π‘ 

πœ†πœ†π‘”π‘”π‘π‘π‘’π‘’π‘”π‘”,𝑠𝑠

2𝐴𝐴𝑠𝑠 π΄π΄π‘ π‘ οΏ½π‘‡π‘‡π‘”π‘”βˆ’π‘‡π‘‡π‘ π‘ οΏ½ (5.49)

where 𝐢𝐢𝑣𝑣 is specific heat of the particle, 𝑁𝑁𝑒𝑒𝑔𝑔,𝑠𝑠 is Nusselt number for heat transfer from gas to the particle. 𝑇𝑇𝑠𝑠 and 𝑇𝑇𝑔𝑔 is the particle and gas temperature respectively.

The chemistry in the CPFD model is specified as mass action kinetics. The chemical reactions are described by stoichiometric equations including the corresponding reaction kinetics. The reaction kinetics is expressed as:

π‘˜π‘˜=𝐴𝐴0π‘šπ‘šπ‘ π‘ π‘π‘1𝑇𝑇𝑐𝑐2exp οΏ½βˆ’ 𝐸𝐸

𝑅𝑅𝑇𝑇+𝐸𝐸0οΏ½ (5.50)

where 𝐴𝐴0 is the pre-exponential factor, 𝐸𝐸 is activation energy, 𝐸𝐸0 is activation energy constant, 𝑅𝑅 is universal gas constant, 𝑝𝑝 is a constant. 𝑇𝑇 is the temperature of a particle gas film.

50 CHAPTER 5 MATHEMATICAL MODEL The film temperature is an average of the particle temperature and the bulk gas temperature. The particle concentration is given by mass per volume and π‘šπ‘šπ‘ π‘  = πœŒπœŒπ‘ π‘ π›Όπ›Όπ‘ π‘ .

The CPFD model is validated against experimental data obtained from bubbling and circulating fluidized bed reactors. The model is used to simulate bubbling and circulating fluidized bed reactors. One of the studies in CFB includes the effect of gas velocity on the bed material outflow at varying bed material feed rates.

The particle out-flow rate vs gas velocity is shown in Figure 5.2. At a given feed rate of particles, the particle outflow rate increases with the gas velocity up to the dimensionless gas velocity about 35. The dimensionless velocity is the ratio of gas velocity to minimum fluidization velocity and when exceeding 35, the solid outflow rate is constant.

Figure 5.2: Solid out-flux vs gas velocity

The velocity range that corresponds to the unsteady outflow rate of the particles should be avoided in order to have a steady state circulation of bed materials. The fluctuation of the bed material outflow is related to the variation of average pressure drop along the height of the riser at that range of gas velocities. The details of the simulation results regarding the flow regime in the CFB combustion reactor and the parameters effected by the flow regimes are discussed in Paper I.

Chapter 6

Biomass properties and reaction kinetics

Characterization of biomass and experimental determination of gasification reaction kinetics are beyond the scope of the present work. The data used in this study are from published literature. The biomass is wood (birch). The wood is considered as a virtual element with the elemental analysis given in Table 6.1 [73].

Table 6.1: Elemental analysis of wood

The table includes only the major components of the wood and the rest of the components are neglected in order to simplify the reactions in the model.

Volatilization of biomass is the first step of the gasification process and it is an important step in the conversion process. In this process most of the wood particles (91 wt.%) are converted to volatiles and tars and the rest is char particles. The composition of the volatiles is presented in Table 6.2. The composition of volatiles given here is in dry basis.

Table 6.2: Composition of volatiles [73]

Components Wt.

fraction Methane (CH4) 0.1213 Carbon monoxide (CO) 0.6856 Carbon-dioxide (CO2 ) 0.1764 Hydrogen (H2) 0.0167

The gasifier in the dual fluidized bed gasification system is operated at a temperature between 800 ̊C and 900 ̊C using pure steam as the gasifying agent.

The conversion reactions in the gasification process are heterogeneous and homogeneous.

Elements Wt.%

Carbon, C 48.6

Hydrogen, H 5.6

Oxygen, O 45.6

Nitrogen, N 0.2

51

52 CHAPTER 6 BIOMASS PROPERTIES AND REACTION KINETICS

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