Trans RINA, Vol 161, Part A4, Intl J Maritime Eng, Oct-Dec 2019


Reaction (1) represents the dissolution of gaseous SO2 in water at the liquid-gas interphase, and it is governed by Henry’s law:


[SO ( )] = SO2 H 2 aq p k (6)


where pSO2 is the partial pressure of SO2, [SO2(aq)] the concentration of SO2 in the solution and kH the Henry’s constant, expressed as (Sander, 2005):


−∆Hsoln 11 k ke HH =


where 0 soln


H 0 RT T 


 −  


0  (7)


k is the Henry’s constant at the reference state, ∆H the enthalpy of solution, T the temperature in


Kelvin and T0 the reference state temperature (298.15 K). A value of 0


kH = 1.2 mol/(kg atm) and the slope -∆H /R soln = 2850 K were employed, Sander (2005).


Reaction (5) represents the dissolution of CO2 at the liquid- gas interphase. This is also governed by Henry’s law.


Reactions (2-4) occur inside the droplet and their kinetics was assumed to be in equilibrium (Chen, et al, 2011 & Andreason,


2007). The equilibrium condition was


imposed by minimizing the Gibbs free energy according to the mass action law. Reactions (2-4) can be written in general form as follows:


aA bB cD dD + ↔+ (8)


where A and B are reactants, C and D products, and a, b, c, and d the stoichiometric coefficients. According to the mass action law, the Gibbs free energy is minimum when:


∆  −=  RT 


G ln 0


[C] [D] [A] [B]


c d ab (9)


where ΔG0 is the standard Gibbs free energy change, R the universal gas constant, [A] the molar concentration of species A at


[C]d[D]d/([A]a[b]b) is the so-called equilibrium constant for the reaction.


2.2 GOVERNING EQUATIONS


The governing equations are the transport equations of conservation of mass, momentum and energy, Eqs. (10- 12) respectively. A laminar simulation was carried out due to the


low droplet diameters and velocities


employed. Inside the droplet, the energy equation was solved until the evaporation temperature was reached. From this temperature, it was imposed as constant and a diameter reduction due to evaporation was implemented in the code.


∇⋅ = 


u Sm (10) where m Hh(T T h and the term h(Te - Tg) =∇ e − g )/ lv


represents the heat transferred between the droplet and gas. Therefore, h(Te - Tg)/hlv is the mass transfer per unit area along


the interface. Te is the evaporation


temperature, Tg the gas temperature, hlv the latent heat of evaporation, and h the heat transfer coefficient, given by the Ranz-Marshall equation (Ranz & Marshall, 1952):


hD + k


= 2 0.6Re Pr 1/2 1/3 (14)


where k is the thermal conductivity, Re the Reynolds number, Pr the Prandtl number and D the droplet diameter. In the present work the shape of


the droplet remained


practically spherical due to the importance of the surface tension force in comparison with the inertial force.


H is the so-called Heaviside function. In the present paper, this takes a 0 value for the liquid, 1 for the vapor and a value between 0 and 1 along the interface, Eq. (15).


equilibrium, and so forth. The term H = 


0 1


  +


if if


 (φ ε ε πφ ε π φ ε ) /(2 ) [sin( / )] /(2 ) if +


φε φ ε ≤


< > where ε is a small quantity of the order of the mesh size.


The source term Se indicated in Eq. (12) also accounts for the phase change, and is given by: e = lv


S hm (16)


The source term Smo of Eq. (11) accounts for the effect of surface tension, given by (Brackbill, et al, 1992):


SH mo = ∇ σκ where κ is the curvature of the interface. (17) (15)


ρρ ρ τ 


∂ + ∇⋅ = −∇ + −∇⋅ + ∂


u t


ρ ρ


c T c uT k T S t


∂ + ∇⋅ = ∇ + ∂


() 


 uu


( )


p g 2


 e Smo


(11) (12)


In the equations above, u is the velocity, ρ the density, t the time, p the pressure, τ the stress tensor, c the specific heat and k the thermal conductivity. The terms Sm, Smo and Se are sources in the equations of mass, momentum and energy respectively.


The source term Sm of Eq. (10) accounts for the phase change, and is given by (Sim, et al, 2015):


Sm ρ ρ


m =  11 − 


vl  (13)


©2019: The Royal Institution of Naval Architects


A-337


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