Advanced Search

Question: Solving a combined system of differential and partially non-linear equations with dsolve/solve

Question:Solving a combined system of differential and partially non-linear equations with dsolve/solve

Durre 10 Maple 18
linear solve diff dsolve nonlinear rootof
September 19 2014
1

Dear Maple enthusiasts,

I am unable to find a working method to solve a system of 8 equations, of which 4 are differential equations. The system contains 8 unknown variables and the goal is to find an expression for each of these variables as a function of the time t. I have attached the code of my project at the bottom of this message.

I have tried the following:

  1. Using solve/dsolve to solve all 8 equations at once. This results in Maple eating up all of my memory and never finishing its calculations.
  2. First using solve to solve the 4 non-differential equations so that I get 4 out of 8 variables as a function of the 4 remaining variables. This results in an expression containing RootOf() for each of the 4 veriables I'm solving for, which prevents me from using these expressions in the 4 remaining differential equations.
  3. First using dsolve to solve the differential equations, which gives once again an expression for 4 variables as a function of the 4 remaining variables. I then use solve to solve the 4 remaining equations with the new found expressions. This results in an extremely long solution for each of the variables.

The code below contains the 3rd option I tried.

Any help or suggestions would be greatly appreciated. I have been scratching my head so much that I'm getting bald and whatever I search for on google or in the Maple help, I can't find a good reference to a system of differential equations together with other equations.

 

 

> 

restart:

PARK - Mixed control
 

 

Input parameters
 

 

Projected interface area (m²)

> 

A_int:=0.025^2*Pi:

 

Temperature of the process (K)

> 

T_proc:=1873:

 

Densities (kg/m³)

> 

Rho_m:=7000: metal
> 

Rho_s:=2850: slag

 

Masses (kg)

> 

W_m:=0.5: metal
> 

W_s:=0.075: slag

 

Mass transfer coefficients (m/s)

> 

m_Al:=3*10^(-4):
> 

m_Si:=3*10^(-4):
> 

m_SiO2:=3*10^(-5):
> 

m_Al2O3:=3*10^(-5):

 

Weight percentages in bulk at t=0 (%)

> 

Pct_Al_b0:=0.3:
> 

Pct_Si_b0:=0:
> 

Pct_SiO2_b0:=5:
> 

Pct_Al2O3_b0:=50:

 

Weight percentages in bulk at equilibrium (%)

> 

Pct_Al_beq:=0.132:
> 

Pct_Si_beq:=0.131:
> 

Pct_SiO2_beq:=3.13:
> 

Pct_Al2O3_beq:=52.12:

 

Weight percentages at the interface (%)

Constants
 

 

Atomic weights (g/mol)

> 

AW_Al:=26.9815385:
> 

AW_Si:=28.085:
> 

AW_O:=15.999:
> 

AW_Mg:=24.305:
> 

AW_Ca:=40.078:

 

Molecular weights (g/mol)

> 

MW_SiO2:=AW_Si+2*AW_O:
> 

MW_Al2O3:=2*AW_Al+3*AW_O:
> 

MW_MgO:=AW_Mg+AW_O:
> 

MW_CaO:=AW_Ca+AW_O:

 

Gas constant (m³*Pa/[K*mol])

> 

R_cst:=8.3144621:

 

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 + (100-Pct_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 + (100-Pct_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^3-a_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);

,


Download Park_-_mixed_control_model.mw
Please Wait...