Euler-Euler method

The Euler-Euler method treats the continuous fluid and dispersed solid as interpenetrating continua. The fluid and solid are treated as primary and secondary phases respectively. The two phases interact with each other by momentum exchange. The model solves a set of conservation equations (e.g. continuity and momentum) for the primary and secondary phases. The secondary phase is differentiated by the solid particle diameter. Each group of particles with a unique diameter is regarded as a separate phase. A single pressure is shared by all the phases. A short description of the method is given in this chapter [63]. The continuity equation for the secondary phase is given by Equation 5.1.

𝜕𝜕

𝜕𝜕𝑝𝑝(𝛼𝛼𝑠𝑠𝜌𝜌𝑠𝑠) +∇ ∙(𝛼𝛼𝑠𝑠𝜌𝜌𝑠𝑠𝑢𝑢�⃗𝒔𝒔) =𝑚𝑚̇𝑔𝑔𝑠𝑠 (5.1) Where 𝑚𝑚̇_{𝑔𝑔𝑠𝑠}is the mass transfer rate, for example due to chemical reaction or evaporation. The granular phase momentum equation is expressed by:

𝜕𝜕

𝜕𝜕𝑝𝑝(𝛼𝛼_𝑠𝑠𝜌𝜌_𝑠𝑠𝑢𝑢�⃗_𝑠𝑠) +∇ ∙(𝛼𝛼_𝑠𝑠𝜌𝜌_𝑠𝑠𝑢𝑢�⃗_𝑠𝑠𝑢𝑢�⃗_𝑠𝑠)

=−𝛼𝛼𝑠𝑠∇𝑃𝑃𝑔𝑔+∇𝜏𝜏𝑠𝑠+��𝑅𝑅�⃗𝑔𝑔𝑠𝑠+𝑚𝑚̇𝑔𝑔𝑠𝑠𝑢𝑢�⃗𝑔𝑔𝑠𝑠�+𝐹𝐹⃗𝑠𝑠 𝑛𝑛

𝑠𝑠=1

(5.2)

39

40 CHAPTER 5. MATHEMATICAL MODEL

In the equation, 𝑃𝑃_𝑔𝑔 is the fluid pressure, 𝜏𝜏_𝑠𝑠 is solid stress and _{𝑅𝑅�⃗𝑔𝑔𝑠𝑠} is the phase interaction term. Since a volume occupied by one phase can not be occupied by another phase, the concept of volume fraction is introduced. The sum of volume fraction of phases equals to one.

𝛼𝛼_𝑠𝑠+𝛼𝛼_𝑔𝑔 = 1 (5.3)

where, 𝛼𝛼_𝑠𝑠in this equation represents the sum of the volume fraction of all possible solid phases whereas 𝛼𝛼𝑔𝑔 is gas phase volume fraction.

The effects of particle-particle interactions are counted for using Kinetic Theory of Granular Flow (KTGF). The KTGF approach was widely accepted as an essential constitutive model for particle flow [64]. The KTGF approach introduces the concept of granular temperature (solid fluctuating energy) of particles. Solid pressure and viscosity are determined by considering the energy dissipation due to particle-particle collision and introducing the concept of the coefficient of restitution [65, 66]. The kinetic energy of particles due to their fluctuating velocity is measured as granular temperature. The granular temperature is proportional to the kinetic energy of the random motion of the particles and is defined as:

𝜃𝜃𝑠𝑠 =1

3〈𝐶𝐶⃗𝑠𝑠𝐶𝐶⃗𝑠𝑠〉 (5.4)

where 𝐶𝐶⃗𝑠𝑠 =𝑢𝑢�⃗𝑠𝑠− 𝑣𝑣⃗𝑠𝑠 is fluctuating velocity of particle and 𝑣𝑣⃗𝑠𝑠 is average particle velocity.

The granular temperature is determined by solving the transport equation, which describes the variation of particle velocity fluctuations. The transport equation is given by:

3

2�𝜕𝜕(𝛼𝛼_𝑠𝑠𝜌𝜌_𝑠𝑠𝜃𝜃_𝑠𝑠)

𝜕𝜕𝑝𝑝 +∇ ∙(𝛼𝛼_𝑠𝑠𝜌𝜌_𝑠𝑠𝑢𝑢�⃗_𝑠𝑠𝜃𝜃_𝑠𝑠)�

=𝜏𝜏̅𝑠𝑠:∇𝑢𝑢�⃗𝑠𝑠+∇ ∙(𝐾𝐾𝜃𝜃𝑠𝑠∇θ𝑠𝑠)− 𝛾𝛾𝑠𝑠+Φ𝑙𝑙𝑚𝑚 +Φ𝑔𝑔𝑠𝑠

(5.5) In the equation above _{𝜏𝜏̅𝑠𝑠:∇𝑢𝑢�⃗𝑠𝑠} is the production of granular temperature by the solid stress,∇ ∙(𝐾𝐾_{𝜃𝜃𝑠𝑠}∇θ𝑠𝑠)is diffusion of granular temperature, 𝐾𝐾𝜃𝜃𝑠𝑠is granulartemperature conductivity, 𝛾𝛾_𝑠𝑠 is dissipation due to particle-particle collision and Φ_{𝑙𝑙𝑚𝑚}+Φ_{𝑔𝑔𝑠𝑠} is the exchange term.

More constitutive equations are needed to account for interphase interaction presented in the moment conservation equation for granular flow (Equation 5.2).

Solid stress ∇𝜏𝜏_𝑠𝑠 accounts for interactions within the particle phase. The solid stress term is derived from the kinetic theory of granular flow and is expressed as:

5.1 EULER – EULER METHOD 41

𝜏𝜏_𝑠𝑠 =−𝑃𝑃_𝑠𝑠𝐼𝐼̅+ 2𝛼𝛼_𝑠𝑠𝜇𝜇_𝑠𝑠𝑆𝑆̅+𝛼𝛼_𝑠𝑠 �𝜆𝜆_𝑠𝑠−2

3𝜇𝜇_𝑠𝑠� ∇ ∙ 𝑢𝑢�⃗_𝑠𝑠𝐼𝐼̅ (5.6)

where

𝑆𝑆⃗=¹₂(∇𝑢𝑢�⃗_𝑠𝑠+ (∇𝑢𝑢�⃗_𝑠𝑠)^𝑇𝑇) = Strain rate 𝑃𝑃_𝑠𝑠 = Solid pressure

𝜆𝜆_𝑠𝑠,𝜇𝜇_𝑠𝑠 = Solid bulk and shear viscosity

Solid pressure is the pressure exerted on the containing wall due to the presence of particles. This is the measure of the momentum transfer due to motion of the particles and collisions. Different models for the solid pressure proposed by different authors are summarized below:

Lun et al.:

𝑃𝑃𝑠𝑠 =𝛼𝛼𝑠𝑠𝜌𝜌𝑠𝑠𝜃𝜃𝑠𝑠+ 2𝜌𝜌𝑠𝑠𝜃𝜃𝑠𝑠(1 +𝑅𝑅𝑠𝑠)𝛼𝛼𝑠𝑠2𝑔𝑔𝑜𝑜𝑠𝑠 (5.7) where 𝛼𝛼_𝑠𝑠𝜌𝜌_𝑠𝑠𝜃𝜃_𝑠𝑠is kinetic contribution and 2𝜌𝜌_𝑠𝑠𝜃𝜃_𝑠𝑠(1 +𝑅𝑅_𝑠𝑠)𝛼𝛼_𝑠𝑠²𝑔𝑔_{𝑜𝑜𝑠𝑠} is collisional contribution.

Syamlal et al.:

𝑃𝑃_𝑠𝑠 = 2𝜌𝜌_𝑠𝑠𝜃𝜃_𝑠𝑠(1 +𝑅𝑅_𝑠𝑠)𝛼𝛼_𝑠𝑠²𝑔𝑔_{𝑜𝑜𝑠𝑠} (5.8) The equation contains only collisional contributions.

Ma and Ahmadi:

𝑃𝑃_𝑠𝑠 =𝛼𝛼_𝑠𝑠𝜌𝜌_𝑠𝑠𝜃𝜃_𝑠𝑠[(1 +𝛼𝛼_𝑠𝑠𝑔𝑔_{𝑜𝑜𝑠𝑠})] +1

2(1 +𝑅𝑅_𝑠𝑠)(1− 𝑅𝑅_𝑠𝑠+ 2𝜇𝜇_{𝑚𝑚𝑡𝑡})] (5.9) 𝜇𝜇𝑚𝑚𝑡𝑡 is frictional viscosity and 𝑅𝑅𝑠𝑠is coefficient of restitution. In the models, 𝑔𝑔0𝑠𝑠(𝛼𝛼_𝑠𝑠) is radial distribution function described as a correction factor that modifies the probability of collision close to packing limit. The expression for the radial distribution function in the Syamlal model is:

𝑔𝑔0𝑠𝑠(𝛼𝛼_𝑠𝑠) = 1

1− 𝛼𝛼𝑠𝑠+ 3𝛼𝛼_𝑠𝑠

2(1− 𝛼𝛼𝑠𝑠)² (5.10)

Solid shear viscosity arises due to translational (kinetic) motion and collisional interaction of particles

𝜇𝜇_𝑠𝑠 = 𝜇𝜇𝑠𝑠,𝑐𝑐𝑜𝑜𝑙𝑙𝑙𝑙 +𝜇𝜇_{𝑠𝑠,𝑘𝑘𝑘𝑘𝑛𝑛} (5.11)

42 CHAPTER 5. MATHEMATICAL MODEL

The collisional contribution to the shear viscosity is given by Lun et al. and adopted by all the other models:

𝜇𝜇𝑠𝑠,𝑐𝑐𝑜𝑜𝑙𝑙𝑙𝑙 =8

5𝛼𝛼_𝑠𝑠²𝜌𝜌_𝑠𝑠𝑑𝑑_𝑠𝑠𝑔𝑔_{𝑜𝑜𝑠𝑠}𝜂𝜂 �𝜃𝜃𝑠𝑠

𝜋𝜋�^1/2

(5.12)

The kinetic term of the shear viscosity is given by the models developed by Syamlal and Gidaspow :

Syamlal:

𝜇𝜇𝑠𝑠,𝑐𝑐𝑜𝑜𝑙𝑙𝑙𝑙 =𝛼𝛼_𝑠𝑠𝜌𝜌_𝑠𝑠𝑑𝑑_𝑠𝑠(𝜃𝜃_𝑠𝑠𝜋𝜋)^1/2

12(2− 𝜂𝜂) �1 +8

5𝜂𝜂(3𝜂𝜂 −2)𝛼𝛼𝑠𝑠𝑔𝑔𝑜𝑜𝑠𝑠� (5.13)

Gidaspow:

𝜇𝜇𝑠𝑠,𝑐𝑐𝑜𝑜𝑙𝑙𝑙𝑙 =5𝜌𝜌𝑠𝑠𝑑𝑑𝑠𝑠(𝜃𝜃_𝑠𝑠𝜋𝜋)^1/2

96𝜂𝜂𝑔𝑔𝑜𝑜𝑠𝑠 �1 +8

5𝜂𝜂𝛼𝛼_𝑠𝑠𝑔𝑔_{𝑜𝑜𝑠𝑠}� (5.14)

The bulk viscosity accounts for particle resistance to expansion and compression, which is given by Lun et al.:

𝜆𝜆𝑠𝑠 =8

3𝛼𝛼𝑠𝑠2𝜌𝜌𝑠𝑠𝑑𝑑𝑠𝑠𝑔𝑔𝑜𝑜𝑠𝑠𝜂𝜂 �𝜃𝜃𝑠𝑠

𝜋𝜋�^1/2 (5.15)

where 𝑑𝑑𝑠𝑠particle diameter.

In the regime of maximum packing (0.63 in ANSYS Fluent) which is also known as frictional packing, the frictional stresses become important. The particles at this stage do not collide but rub against each other. Therefore, the momentum transfer occurs through friction. The granular flow becomes incompressible. The packing limit is the maximum limit of granular volume fraction in a bed. The frictional stresses are determined from soil mechanics [67]:

𝜇𝜇𝑠𝑠,𝑚𝑚𝑡𝑡𝑘𝑘𝑐𝑐𝑡𝑡 =𝑃𝑃_𝑠𝑠 𝑠𝑠𝑝𝑝𝑑𝑑𝑠𝑠 2�𝐼𝐼2

(5.16) The effective frictional viscosity in the granular phase is determined from the maximum of the frictional and shear viscosities.

𝜇𝜇_𝑠𝑠 =𝑚𝑚𝑝𝑝𝑚𝑚 �𝜇𝜇𝑠𝑠,𝑐𝑐𝑜𝑜𝑙𝑙𝑙𝑙 +𝜇𝜇_{𝑠𝑠,𝑘𝑘𝑘𝑘𝑛𝑛},𝜇𝜇_{𝑚𝑚𝑡𝑡𝑘𝑘𝑐𝑐𝑡𝑡}� (5.17) Interaction between phases is based on forces on a single particle corrected for effects such as concentration, clustering particles shape and mass transfer effects.

The sum of all forces vanishes:

5.1 EULER – EULER METHOD 43

��𝑅𝑅�⃗_{𝑔𝑔𝑠𝑠}+𝑚𝑚̇𝑔𝑔𝑠𝑠𝑢𝑢�⃗_{𝑔𝑔𝑠𝑠}�= 0

𝑛𝑛 𝑠𝑠=1

(5.18)

Drag is a force caused by relative motion between phases.

� �𝐾𝐾_{𝑙𝑙𝑠𝑠}(𝑢𝑢�⃗_𝑙𝑙− 𝑢𝑢�⃗_𝑠𝑠)) +𝐾𝐾_{𝑔𝑔𝑠𝑠}�𝑢𝑢�⃗_𝑔𝑔− 𝑢𝑢�⃗_𝑠𝑠��= 0

𝑛𝑛 𝑙𝑙=1

(5.19) Where 𝐾𝐾𝑔𝑔𝑠𝑠is the drag between fluid and particles and 𝐾𝐾𝑙𝑙𝑠𝑠 is drag between particles.

The general form of drag term is given by:

𝐾𝐾𝑔𝑔𝑠𝑠 =𝛼𝛼𝑠𝑠𝜌𝜌𝑠𝑠𝑓𝑓_{𝑑𝑑𝑡𝑡𝑑𝑑𝑔𝑔}

𝜏𝜏𝑔𝑔𝑠𝑠 (5.20)

With particle relaxation time 𝜏𝜏𝑔𝑔𝑠𝑠 = 𝜌𝜌_𝑠𝑠𝑑𝑑_𝑠𝑠²

18𝜇𝜇_𝑔𝑔 (5.21)

The Syamlal & O’Brien drag model is used in granular flows to compute the drag forces between fluid and solid phases.

𝑘𝑘_{𝑔𝑔𝑠𝑠} =4

3𝐶𝐶_𝐷𝐷�𝛼𝛼_𝑠𝑠𝛼𝛼_𝑔𝑔𝜌𝜌_{𝑔𝑔�𝜈𝜈𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠�} 𝑑𝑑_𝑠𝑠 �𝑅𝑅𝑅𝑅_𝑠𝑠

𝜈𝜈_𝑡𝑡³ (5.22)

𝐶𝐶_𝐷𝐷 =�0.63 + 4.8

�𝑅𝑅𝑅𝑅/𝜈𝜈𝑡𝑡�

2

(5.23)

𝜈𝜈𝑡𝑡 = 0.5�𝐴𝐴 −0.06𝑅𝑅𝑅𝑅

+�(0.006𝑅𝑅𝑅𝑅)²+ 0.12𝑅𝑅𝑅𝑅(2𝐵𝐵 − 𝐴𝐴) +𝐴𝐴²� (5.24)

𝐴𝐴 =𝛼𝛼_𝑔𝑔^4.41

𝐵𝐵=�0.8𝛼𝛼𝑔𝑔1.28 𝑓𝑓𝑓𝑓𝐴𝐴 𝛼𝛼𝑔𝑔 ≤0.85 0.8𝛼𝛼_𝑔𝑔^2.65 𝑓𝑓𝑓𝑓𝐴𝐴 𝛼𝛼_𝑔𝑔 ≥0.85

(5.25)

Experiments were performed with glass particles and air as fluidizing gas in the cold model of bubbling fluidized bed. The glass particles have about the same size and density as the bed materials used in the dual fluidized biomass gasification

44 CHAPTER 5. MATHEMATICAL MODEL reactor. A series of simulations were run using different drag models to find which model gives the best results. The drag and granular viscosity is calculated using Syamlal-O’Brien model. Frictional viscosity is calculated using Schaeffer model whereas granular bulk viscosity is kept constant. Radial distribution function and solid pressure are calculated using the Ma-Ahmadi model. The validated CFD model is used to study the flow behavior in the cold model of bubbling fluidized bed gasification reactor.

The model is used to investigate Glicksman’s dimensionless scaling parameters.

A ‘reference’ bed and a ‘scaled’ bed are simulated using Glicksman’s full set and simplified sets of dimensionless scaling parameters. In the bubbling fluidized bed gasification reactor, olivine or silica sand particles are used as bed materials with the high temperature steam as fluidizing gas. Down scaling, the reactor using Glicksman’s rule to use ambient air needs particles with density of about 12000 kg/m³. Consequently, it is difficult to verify Glicksman’s scaling rule experimentally for the biomass gasification reactor. This difficulties are easy to overcome with the CFD model and simulated results in this work.

The CFD model is also used to verify Glicksman’s viscous limit set of dimensionless parameters. The viscous limit set is more flexible for scaling of gasification reactors.

A reference bed with lower particle Reynold’s number is scaled down applying Glicksman’s viscous limit sets of dimensionless parameters. The fluid dynamic properties such as pressure fluctuations and solid volume fraction fluctuations are monitored at a number of equally distributed locations in the beds. The pressure fluctuation and the solid volume fraction fluctuations are similar for the reference and scaled beds at particle Reynolds number up to 15. The solid volume fraction fluctuation of the two beds as a function of time is presented in Figure 5.1. The figure shows the similarity in particle flow between the two beds.

The results confirm that fluidized beds with smaller particle size and operating at low gas velocities can be scaled by using the viscous limit set of dimensionless parameters.

In document Optimization of flow behavior in biomass gasification reactor (sider 59-65)