Heat and mass transfer model

Chapter 6 Macroscopic modeling of the dominant size enlargement mechanism

0.15 0.2 0.25 0.4

0 0.2 0.4 0.6 0.8 1 1.2

0.12

0.56

0.94 1

0.12

0.56

0.94 1

𝑥_crit [mm]

𝑃coll,wet,suc[−]

MCanalytical

Figure 6.3:Comparison of the probability of successful wet collisions obtained with the proposed analytical model and a Monte Carlo model.

6.1 Model description 𝑀¤_evap 𝐻¤_evap

𝑄¤_gp 𝑄¤_pl

𝑄¤_gl

Figure 6.4:Schematic representation of the considered mass, enthalpy, and heat flow rates in the presented model.

equal to its maximum. Correspondingly, the evaporation rate lies between zero and the maximum for 0≤𝛹_wet ≤ 1. Changes in the wet surface area can be attributed to changing wet surface fraction due to drying and changing total particle surface area due to size enlargement:

d𝐴_pl d𝑡 = d

d𝑡 𝛹_wet𝐴_p,tot

=𝛹_wetd𝐴_p,tot

d𝑡 +𝐴_p,totd𝛹_wet

d𝑡 . (6.11)

The change of total surface area can be calculated from the transient behavior of the second moment of the particle size distribution𝑛:

𝛹_wetd𝐴_p,tot

d𝑡 =𝛹_wet𝜋

∫∞ 0

𝑥²𝜕𝑛

𝜕𝑡 d𝑥. (6.12)

In contrast to Heinrich and Mörl [141], who assume a coherent film, the liquid phase is here described by a number of individual droplets deposited on the particle surface. Similar to the presented Monte Carlo models, the droplets are assumed to be monodisperse, each covering a certain surface area of the particle. Coalescence or overlapping of droplets are not taken into account. Using these assumptions, the wet surface area can be described based on the contact area between a deposited droplet and the particle𝐴_contact(footprint), the droplet mass𝑀_drop, and the liquid mass𝑀_l:

d𝐴_pl

d𝑡 = 𝐴_contact 𝑀_drop

d𝑀_l

d𝑡 . (6.13)

The transient behavior of the wet surface fraction can then be written as follows, assuming that it does not exceed unity:

d𝛹_wet d𝑡 =





1 𝐴_p,tot





𝐴_contact 𝑀_drop

d𝑀_l

d𝑡 −𝛹_wet𝜋

∫∞ 0

𝑥²𝜕𝑛

𝜕𝑡 d𝑥

 𝛹_wet <1,

0 otherwise.

(6.14)

Chapter 6 Macroscopic modeling of the dominant size enlargement mechanism

The geometry of a deposited droplet is assumed to be a spherical cap, described by the model proposed by Meric and Erbil [151]. The contact area between the droplet and the particle required in Equation (6.14) is calculated using Equation (3.12). Since this parameter depends on the droplet size (volume) and the contact angle, the influence of the wetting parameters on the wet surface fraction and therefore the dominant size enlargement mechanism can be taken into account. In this study, shrinkage and an increasing viscosity of deposited droplets during drying is not considered. The droplet properties are calculated using the initial droplet volume, which follows from the droplet diameter. Extensions in this direction can be made without conceptional difficulties.

Mass and enthalpy balances

The water mass and enthalpy balances for the gas phase are given in Equation (6.15) and Equa-tion (6.16), respectively. Since the gas phase is modeled assuming plug flow, the water mass and enthalpy depend on the spatial location in the fluidized bed, represented by a normalized height coordinate𝜁. Both the water mass and enthalpy are transported along𝜁with flow rates equal to𝑀¤_w,g and𝐻¤_g, respectively. Furthermore, the water mass changes due to the mass flow rate of evaporation 𝑀¤_evapand the enthalpy changes due to the enthalpy flow rate of evaporation𝐻¤_evap, the heat flow rates between the gas and particle phase𝑄¤_gp, and the gas and liquid phase𝑄¤_gl:

𝜕𝑀_w,g

𝜕𝑡 =−𝜕𝑀¤_w,g

𝜕𝜁 + ¤𝑀_evap, (6.15)

𝜕𝐻_g

𝜕𝑡 =−𝜕𝐻¤_g

𝜕𝜁 + ¤𝐻_evap− ¤𝑄_gp− ¤𝑄_gl. (6.16)

In the presented model, the sprayed material is instantly distributed between the film phase and the particle phase. This means that the liquid part of the sprayed material is added to the film phase and the solid part is added to the particle phase. In this way, the kinetics of the underlying process (e.g., crystallization or precipitation) is not explicitly modeled. Then, the mass of the liquid film phase𝑀_l depends on the liquid part of the spraying rate and the mass flow rate of evaporation. The liquid film enthalpy𝐻_lchanges due to evaporation and the heat flow rates between the liquid phase and the gas and particle phase, respectively. Since the liquid phase is assumed to be perfectly mixed, no dependency on the spatial location in the fluidized bed needs to be considered:

d𝑀_l

d𝑡 =𝑀¤_spray,l − ¤𝑀_evap, (6.17)

d𝐻_l

d𝑡 = − ¤𝐻_evap+ ¤𝑄_gl+ ¤𝑄_pl. (6.18)

The mass of the particle phase𝑀_pcan be calculated from the third moment of the particle size distribution. The enthalpy𝐻_pdepends on the heat flow rates between the particle phase and the gas and liquid phase, respectively. The mass and enthalpy of the particle phase are independent of the

6.1 Model description

spatial location as well:

𝑀_p= 𝜋

6𝜚_p𝜇₃= 𝜋 6𝜚_p

∫∞ 0

𝑥³𝑛d𝑥, (6.19)

d𝐻_p

d𝑡 =𝑄¤_gp− ¤𝑄_pl. (6.20)

Since the mass flow rate of evaporation, the corresponding enthalpy flow rate and the heat flow rates between the gas phase and the liquid phase, and between the gas phase and the particle phase depend on the spatial location𝜁, their average values are used in the above shown mass and enthalpy balances for the film and particle phase. The averaged values are calculated as follows:

𝑀¤_evap=

∫ 𝑀¤_evapd𝜁 , (6.21)

𝐻¤_evap =

∫ 𝐻¤_evapd𝜁 , (6.22)

𝑄¤_gl =

∫ 𝑄¤_gld𝜁 , (6.23)

𝑄¤_gp=

∫ 𝑄¤_gpd𝜁 . (6.24)

Kinetics

The mass flow rate of evaporation is calculated using the following equation:

𝑀¤_evap=𝛽𝜚_g𝐴_gl(𝑌_sat(𝑇_l) −𝑌) with 𝐴_gl =𝛹_wet 𝐴_drop

𝐴_contact𝐴_p,tot. (6.25)

In this equation,𝛽is the mass transfer coefficient calculated according to Groenewold and Tsotsas [143] as shown in Appendix B,𝜚_gis the density of the fluidization gas,𝐴_glis the gas liquid interface (curved droplet surface area),𝑌_sat(𝑇_l)is the saturation moisture content of the fluidization gas at liquid film temperature𝑇_lcalculated using Equation (A.11), and𝑌 is the moisture content of the bulk gas. The gas liquid interface is calculated using the wet surface fraction, the total particle surface area and the ratio of the curved droplet surface area and the contact area. In contrast to Heinrich and Mörl [141], the curved droplet surface area and the contact area are not identical in the present approach due to the used droplet geometry model. As a result, the ratio of𝐴_dropand𝐴_contactmust be taken into account when calculating the area of the interface between gas and liquid. Equation (6.25) shows that the wet surface fraction directly influences the mass flow rate of evaporation. The resulting drying rate reaches its maximum if𝛹_wet →1 and goes to zero if𝛹_wet →0, resembling the behavior of particles which first dry from their surface and then from their interior. The curved surface area of the droplet𝐴_dropis calculated using Equation (3.9).

The moisture content of the bulk gas is calculated from the water mass in the gas phase and the mass

Chapter 6 Macroscopic modeling of the dominant size enlargement mechanism

of dry gas in the fluidized bed:

𝑌 = 𝑀_w,g

𝑀_g,dry with 𝑀_g,dry =𝜀_bed𝜋

4𝑑_bed² 𝜚_gℎ_bed. (6.26)

The calculation of the porosity and the height of the bed is performed as shown in Appendix B. The bed diameter𝑑_bedis given by the diameter of the fluidized bed chamber. Similar to Equation (6.26), the moisture content of the particles is calculated using the liquid film mass and the dry bed mass given by the mass of the particles:

𝑋 = 𝑀_l

𝑀_p. (6.27)

The enthalpy flow rate used in the differential equations shown above is calculated as follows:

𝐻¤_evap=𝑀¤_evap 𝑐_v𝑇_l +Δℎ_evap

. (6.28)

The heat flow rates between the respective phases are calculated using the following equations:

𝑄¤_gl =𝛼_gl𝐴_gl 𝑇_g −𝑇_l, (6.29)

𝑄¤_pl =𝛼_pl𝐴_pl 𝑇_p−𝑇_l with 𝐴_pl =𝛹_wet𝐴_p,tot, (6.30)

𝑄¤_gp=𝛼_gp𝐴_gp 𝑇_g −𝑇_p with 𝐴_gp=(1−𝛹_wet)𝐴_p,tot. (6.31) In these equations,𝛼is the heat transfer coefficient between the respective phases. The heat transfer coefficient for the gas-particle heat transfer𝛼_gpis calculated according to Groenewold and Tsotsas [143] as shown in Appendix B. Following Heinrich and Mörl [141], the gas-liquid heat transfer coefficient𝛼_glis assumed to be equal to𝛼_gp. Heat transfer between particle and liquid is assumed to be purely conductive, neglecting any convection. For this special case (spherical particle in contact with a fluid),𝛼_plcan be calculated using a Nusselt number equal to two. The interfaces between the particle and liquid phase𝐴_pland the gas and particle phase𝐴_gprepresent the wet surface area and the dry surface area, respectively. They are calculated as shown in Equation (6.30) and Equation (6.31).

The following equations are used to relate the temperature of each phase with the corresponding enthalpy:

𝐻_g=𝑀_g,dry 𝑐_g𝑇_g+𝑌 𝑐_v𝑇_g+Δℎevap

, (6.32)

𝐻_p=𝑀_p𝑐_p𝑇_p, (6.33)

𝐻_l =𝑀_l𝑐_l𝑇_l. (6.34)

In these equations,𝑐_g,𝑐_v,𝑐_p, and𝑐_lare the specific heat capacities of the gas phase (air), vapor, particle phase, and the liquid phase (water), respectively. The calculation of the specific heat capacities of air, vapor, and water is shown in Appendix A. The specific heat capacity of the particles is given in the corresponding tables along with further simulation parameters.

6.1 Model description

Im Dokument Microscopic and macroscopic modeling of particle formation processes in spray fluidized beds (Seite 132-137)