
PARK  Mixed control



Input parameters


Projected interface area (m²)
Temperature of the process (K)
Densities (kg/m³)
Masses (kg)
Mass transfer coefficients (m/s)
Weight percentages in bulk at t=0 (%)
Weight percentages in bulk at equilibrium (%)
Weight percentages at the interface (%)


Constants


Atomic weights (g/mol)
Molecular weights (g/mol)
> 
MW_Al2O3:=2*AW_Al+3*AW_O:

Gas constant (m³*Pa/[K*mol])


Variables


> 
with(PDEtools): declare((Pct_Al_b(t),Pct_Al_i(t),Pct_Si_b(t),Pct_Si_i(t),Pct_SiO2_b(t),Pct_SiO2_i(t),Pct_Al2O3_b(t),Pct_Al2O3_i(t))(t),prime=t):



Equations



4 rate equations


> 
Rate_eq1:=diff(Pct_Al_b(t),t)=A_int*Rho_m*m_Al/W_m*(Pct_Al_b(t)Pct_Al_i(t));

> 
Rate_eq2:=diff(Pct_Si_b(t),t)=A_int*Rho_m*m_Si/W_m*(Pct_Si_b(t)Pct_Si_i(t));

> 
Rate_eq3:=diff(Pct_SiO2_b(t),t)=A_int*Rho_s*m_SiO2/W_s*(Pct_SiO2_b(t)Pct_SiO2_i(t));

> 
Rate_eq4:=diff(Pct_Al2O3_b(t),t)=A_int*Rho_s*m_Al2O3/W_s*(Pct_Al2O3_b(t)Pct_Al2O3_i(t));



3 mass balance equations


> 
Mass_eq1:=0=(Pct_Al_b(t)Pct_Al_i(t))+4*AW_Al/(3*AW_Si)*(Pct_Si_b(t)Pct_Si_i(t));

> 
Mass_eq2:=0=(Pct_Al_b(t)Pct_Al_i(t))+4*Rho_s*m_SiO2*W_m*AW_Al/(3*Rho_m*m_Al*W_s*MW_SiO2)*(Pct_SiO2_b(t)Pct_SiO2_i(t));

> 
Mass_eq3:=0=(Pct_Al_b(t)Pct_Al_i(t))+2*Rho_s*m_Al2O3*W_m*AW_Al/(Rho_m*m_Al*W_s*MW_Al2O3)*(Pct_Al2O3_b(t)Pct_Al2O3_i(t));



1 local equilibrium equation


Gibbs free energy of the reaction when all of the reactants and products are in their standard states (J/mol). Al and Si activities are in 1 wt pct standard state in liquid Fe. SiO2 and Al2O3 activities are in respect to pure solid state.
> 
delta_G0:=720680+133*T_proc:

Expression of mole fractions as a function of weight percentages (whereby MgO is not taken into account, but instead replaced by CaO ?)
> 
x_Al2O3_i(t):=(Pct_Al2O3_i(t)/MW_Al2O3)/(Pct_Al2O3_i(t)/MW_Al2O3 + Pct_SiO2_i(t)/MW_SiO2 + (100Pct_SiO2_i(t)Pct_Al2O3_i(t))/MW_CaO); x_SiO2_i(t):=(Pct_SiO2_i(t)/MW_SiO2)/(Pct_Al2O3_i(t)/MW_Al2O3 + Pct_SiO2_i(t)/MW_SiO2 + (100Pct_SiO2_i(t)Pct_Al2O3_i(t))/MW_CaO);

Activity coefficients
> 
Gamma_Al_Hry:=1: because very low percentage present during the process (~Henry's law)

> 
Gamma_Si_Hry:=1: because very low percentage present during the process (~Henry's law)

> 
Gamma_Al2O3_Ra:=1: temporary value!

> 
Gamma_SiO2_Ra:=10^(4.85279678314968+0.457486603678622*Pct_SiO2_b(t)); very small activity coefficient? plot(10^(4.85279678314968+0.457486603678622*Pct_SiO2_b),Pct_SiO2_b=3..7);

Activities of components
> 
a_Al_Hry:=Gamma_Al_Hry*Pct_Al_i(t); a_Si_Hry:=Gamma_Si_Hry*Pct_Si_i(t); a_Al2O3_Ra:=Gamma_Al2O3_Ra*x_Al2O3_i(t); a_SiO2_Ra:=Gamma_SiO2_Ra*x_SiO2_i(t);

Expressions for the equilibrium constant K
> 
K_cst:=exp(delta_G0/(R_cst*T_proc));

> 
Equil_eq:=0=K_cst*a_Al_Hry^4*a_SiO2_Ra^3a_Si_Hry^3*a_Al2O3_Ra^2;




Output


> 
with(ListTools): dsys:=Rate_eq1,Rate_eq2,Rate_eq3,Rate_eq4: dvars:={Pct_Al2O3_b(t),Pct_SiO2_b(t),Pct_Al_b(t),Pct_Si_b(t)}: dconds:=Pct_Al2O3_b(0)=Pct_Al2O3_b0,Pct_SiO2_b(0)=Pct_SiO2_b0,Pct_Si_b(0)=Pct_Si_b0,Pct_Al_b(0)=Pct_Al_b0: dsol:=dsolve({dsys,dconds},dvars):

> 
Pct_Al2O3_b(t):=rhs(select(has,dsol,Pct_Al2O3_b)[1]); Pct_Al_b(t):=rhs(select(has,dsol,Pct_Al_b)[1]); Pct_SiO2_b(t):=rhs(select(has,dsol,Pct_SiO2_b)[1]); Pct_Si_b(t):=rhs(select(has,dsol,Pct_Si_b)[1]);

> 
sys:={Equil_eq,Mass_eq1,Mass_eq2,Mass_eq3}: vars:={Pct_Al2O3_i(t),Pct_SiO2_i(t),Pct_Al_i(t),Pct_Si_i(t)}: sol:=solve(sys,vars);



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