lsqcurvefit gives an error message:“Index exceeds array bounds.”

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Dursman Mchabe
Dursman Mchabe 2018-10-1
I have a set of 7 ordinary differential equations and 1 algebraic which I use to calculate the concentrations of 8 species with time. I have also conducted experiments to measure the concentrations of such 8 species. The challenge I have now that the concentrations predicted by the equations are not the same as the concentrations measured during experiments. As a result, I am looking at ways the regress the 4 parameters and 8 initial conditions, such that the calculated concentrations can match the experimental concentrations. I have found an example on the internet, from this link:
https://www.mathworks.com/matlabcentral/answers/329901-solving-coupled-differential-equations
in which the approach of using matlab's lsqcurvefit was followed. I Have tried to follow the step by step, however, I get the error message:
Index exceeds array bounds.
Error in sym/subsref (line 859)
R_tilde = builtin('subsref',L_tilde,Idx);
Error in EstimationOfParametersAndInitialConditionsUsingLSQCURVEFIT3 (line 73)
eqn1 = diff(b(5), t)== F_rate/ V_Headspace * ( CSO2_in - b(5)) - (b(5) * R * T - HSO2 * ((b(7) *
b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) / ((1/b(1)) + 1/(b(2) * (((DSO2 * (HSO2 * b(5) * R * T -
((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) + DHSO3 * (((KSO2 * HSO2 * b(5) * R * T / b(12)) -
((b(7) * KSO2 * b(12))/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))) + DSO32 * ((((KSO2 * KHSO3 * HSO2 * b(5) * R *
T/b(12)^2) - ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))))) / (DSO2 * (HSO2 * b(5) * R * T
- ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))))))));
My current code is:
function EstimationOfParametersAndInitialConditionsUsingLSQCURVEFIT3
global ExperimentalTime ExperimentalConcentrations
ExperimentalTime = [600
1200
1800
2400
3000
10200
17400
24600
31800
39000
46200
53400
60600
67800
75000
82200
89400
96600
103800
111000
118200
125400
132600
139800
147000
154200
161400
168600
175800
183000];
ExperimentalConcentrations = [ 0.00E+00 6.97E-08 3.34E-01 4.88E-06 4.88E-06 4.86E+01 0.00E+00 7.41E-06
1.72E-02 8.76E-08 6.79E-01 4.88E-06 4.88E-06 4.92E+01 0.00E+00 9.33E-06
1.71E-02 1.18E-07 1.04E+00 5.11E-06 5.11E-06 4.94E+01 0.00E+00 1.20E-05
1.70E-02 1.03E-07 1.41E+00 5.11E-06 5.11E-06 4.96E+01 0.00E+00 1.05E-05
1.68E-02 1.77E-07 6.00E+00 5.35E-06 5.35E-06 3.78E+01 6.30E+00 1.74E-05
1.74E-02 3.65E-07 1.07E+01 5.35E-06 5.35E-06 3.09E+01 9.33E+00 3.72E-05
1.65E-02 3.50E-07 1.56E+01 5.36E-06 5.36E-06 2.67E+01 1.13E+01 3.55E-05
1.67E-02 8.83E-07 2.07E+01 5.36E-06 5.36E-06 1.98E+01 1.43E+01 1.00E-04
1.67E-02 5.01E-06 2.59E+01 5.07E-06 5.07E-06 1.14E+01 1.79E+01 4.07E-02
1.70E-02 5.06E-06 3.14E+01 5.07E-06 5.07E-06 9.58E+00 1.91E+01 2.45E-01
1.80E-02 4.93E-06 3.67E+01 4.93E-06 4.93E-06 5.65E+00 2.09E+01 6.17E-01
2.15E-02 5.29E-06 4.18E+01 5.30E-06 5.30E-06 3.32E+00 2.21E+01 1.32E+00
2.68E-02 5.30E-06 4.66E+01 5.30E-06 5.30E-06 0.00E+00 2.37E+01 2.29E+00
3.07E-02 5.30E-06 5.09E+01 5.30E-06 5.30E-06 0.00E+00 2.40E+01 2.34E+00
3.33E-02 9.45E-06 5.51E+01 9.46E-06 9.46E-06 0.00E+00 2.43E+01 2.40E+00
3.68E-02 9.45E-06 5.90E+01 9.46E-06 9.46E-06 0.00E+00 2.47E+01 1.82E+00
4.14E-02 1.13E-05 6.23E+01 1.13E-05 1.13E-05 0.00E+00 2.50E+01 1.38E+00
4.48E-02 1.15E-05 6.56E+01 1.15E-05 1.15E-05 0.00E+00 2.54E+01 2.09E+00
4.87E-02 1.10E-05 6.87E+01 1.10E-05 1.10E-05 0.00E+00 2.58E+01 1.82E+00
5.19E-02 1.10E-05 7.12E+01 1.10E-05 1.10E-05 0.00E+00 2.62E+01 1.58E+00
5.47E-02 1.04E-05 7.36E+01 1.04E-05 1.04E-05 0.00E+00 2.67E+01 2.29E+00
5.75E-02 1.04E-05 7.59E+01 1.04E-05 1.04E-05 0.00E+00 2.71E+01 1.62E+00
5.97E-02 1.04E-05 7.78E+01 1.04E-05 1.04E-05 0.00E+00 2.76E+01 1.12E+00
6.17E-02 1.03E-05 7.95E+01 1.03E-05 1.03E-05 0.00E+00 2.81E+01 8.91E-01
6.33E-02 1.04E-05 8.11E+01 1.04E-05 1.04E-05 0.00E+00 2.86E+01 8.51E-01
6.44E-02 1.05E-05 8.24E+01 1.06E-05 1.06E-05 0.00E+00 2.90E+01 7.59E-01
6.48E-02 1.13E-05 8.24E+01 1.13E-05 1.13E-05 0.00E+00 2.90E+01 8.71E-01
6.53E-02 1.13E-05 8.23E+01 1.13E-05 1.13E-05 0.00E+00 2.90E+01 1.12E+00
6.57E-02 1.13E-05 8.21E+01 1.13E-05 1.13E-05 0.00E+00 2.90E+01 1.00E+00
6.59E-02 1.03E-05 8.20E+01 1.03E-05 1.03E-05 0.00E+00 2.90E+01 8.51E-01];
%-------------------------------------------------------------------------------------------------------------------------------------
% b(1:4) = Parameters, b(5:12) = Initial Conditions
% MAPPING: b(1) = kga, b(2) = kLa_SO2, b(3) = ktot, b(4) = Kad,
% b(5)= CSO2_gas(t0), b(6)= CCO2_gas(t0), b(7)= S_total(t0), b(8)= C_total(t0), b(9)= Ca2_total(t0),b(10)= CCaCO3(t0),b(11)= CCaSO3(t0),b(12)= CH(t0)
syms b V_Headspace F_rate CSO2_in CCO2_in R T HSO2 DCa2 DSO2 DHSO3 DSO32 kLa_CO2 HCO2 DCO2 DHCO3 DCO32 KSPCaSO3 BETCaSO3 BETCaCO3 MWCaCO3 KCO2 KHCO3 KSO2 KHSO3 Kw CCa2
eqn1 = diff(b(5), t)== F_rate/ V_Headspace * ( CSO2_in - b(5)) - (b(5) * R * T - HSO2 * ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) / ((1/b(1)) + 1/(b(2) * (((DSO2 * (HSO2 * b(5) * R * T - ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) + DHSO3 * (((KSO2 * HSO2 * b(5) * R * T / b(12)) - ((b(7) * KSO2 * b(12))/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))) + DSO32 * ((((KSO2 * KHSO3 * HSO2 * b(5) * R * T/b(12)^2) - ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))))) / (DSO2 * (HSO2 * b(5) * R * T - ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))))))));
eqn2 = diff(b(6), t)== F_rate/ V_Headspace * ( CCO2_in - b(6)) - (kLa_CO2 * ((((DCO2 * (HCO2 * b(6) * R * T - ((b(8) * b(12)^2)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3))) + DHCO3 *(((KCO2 * HCO2 * b(6) * R * T/b(12)) - ((C_total (t) * KCO2 * b(12))/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3)))) + DCO32 * ((((KCO2 * KHCO3 * HCO2 * b(6) * R * T/b(12)^2) - ((b(8) * KCO2 * KHCO3)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3)))))) / (DCO2 * (HCO2 * b(6) * R * T - ((C_total (t) * b(12)^2)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3)))))))*(b(6) * R * T/HCO2 - ((b(8) * b(12)^2)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3))));
eqn3 = diff(b(7), t)== (b(5) * R * T - HSO2 * ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) / ((1/b(1)) + 1/(b(2) * (((DSO2*(HSO2* b(5) * R * T - ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) + DHSO3*(((KSO2 * HSO2 * b(5) * R * T/b(12)) - ((b(7) * KSO2 * b(12))/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))) + DSO32 * ((((KSO2 * KHSO3 *HSO2* b(5) * R * T/b(12)^2) - ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))))) / (DSO2 * (HSO2 * b(5) * R * T - ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))))))) - 0.162 * exp(-5153/T) * (((CCa2 * ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) / KSPCaSO3) - 1)^2 * (BETCaSO3/((CCa2 * (((((KSO2 * KHSO3 *HSO2* b(5) * R * T/b(12)^2) - ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))))) / (DSO2 * (HSO2 * b(5) * R * T - ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))))))))/ KSPCaSO3;
eqn4 = diff(b(8), t)== (kLa_CO2 * ((((DCO2 * (HCO2 * b(6) * R * T - ((b(8) * b(12)^2)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3))) + DHCO3 * (((KCO2 * HCO2 * b(6) * R * T/b(12)) - ((C_total (t) * KCO2 * b(12))/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3)))) + DCO32 * ((((KCO2 * KHCO3 * HCO2 * b(6) * R * T/b(12)^2) - ((b(8) * KCO2 * KHCO3)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3)))))) / (DCO2 * (HCO2 * b(6) * R * T - ((C_total (t) * b(12)^2)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3)))))))*(b(6) * R * T /HCO2 - ((b(8) * b(12)^2)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3)))) + (b(3) * BETCaCO3 * MWCaCO3 * b(10) * b(12) *(1 - (b(4) * b(12))/(1 + b(4) * b(12))));
eqn5 = diff(b(9), t)== (-1) * (b(3) * BETCaCO3 * MWCaCO3 * b(10) * b(12) *(1 - (b(4) * b(12))/(1 + b(4) * b(12)))) - (0.162 * exp(-5153/T) * (((CCa2 * ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) / KSPCaSO3) - 1)^2 * (BETCaSO3/((CCa2 * (((((KSO2 * KHSO3 *HSO2* b(5) * R * T/b(12)^2) - ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))))) / (DSO2 * (HSO2 * b(5) * R * T - ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))))))))/ KSPCaSO3;
eqn6 = diff(b(10), t)== (-1) * (b(3) * BETCaCO3 * MWCaCO3 * b(10) * b(12) * (1 - (b(4) * b(12))/(1 + b(4) * b(12))));
eqn7 = diff(b(11), t)== 0.162 * exp(-5153/T) * ((((CCa2 * ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))) / KSPCaSO3) - 1)^2 * (BETCaSO3/((CCa2 * (((((KSO2 * KHSO3 * HSO2 * b(5) * R * T/b(12)^2) - ((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)))))) / (DSO2 * (HSO2 * b(5) * R * T - ((b(7) * b(12)^2)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))))))))/ KSPCaSO3;
eqn8 = diff(b(12), t)== b(12) + 2 * CCa2 - ((b(7) * KSO2 * b(12))/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3))- 2*((b(7) * KSO2 * KHSO3)/(b(12)^2 + KSO2 * b(12) + KSO2 * KHSO3)) - ((b(8) * KCO2 * b(12))/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3)) - 2 * ((b(8) * KCO2 * KHCO3)/(b(12)^2 + KCO2 * b(12) + KCO2 * KHCO3))- Kw / b(12);
eqns = [eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8];
vars = [b(5); b(6); b(7); b(8); b(9); b(10); b(11); b(12)];
origVars = length(vars);
M = incidenceMatrix(eqns, vars);
[eqns, vars] = reduceDifferentialOrder(eqns, vars);
isLowIndexDAE(eqns,vars);
f = daeFunction(eqns,vars,b, V_Headspace, F_rate, CSO2_in, CCO2_in, CCa2, R, T, HSO2, DCa2, DSO2, DHSO3, DSO32, kLa_CO2, HCO2, DCO2, DHCO3, DCO32, KSPCaSO3, BETCaSO3, BETCaCO3, MWCaCO3, KCO2, KHCO3, KSO2, KHSO3, Kw);
V_Headspace = 1.5e-6;
F_rate = 1.66667e-5;
CSO2_in = 6.51332e-2;
CCO2_in = 0;
CCa2 = 10;
R = 8.314;
T = 323.15;
HSO2 = 149;
b(1) = 4.14e-6;
DCa2 = 1.39e-9;
DSO2 = 2.89e-9;
DHSO3 = 2.89e-9;
DSO32 = 2.89e-9;
b(2) = 8.4e-4;
kLa_CO2 = 9.598e-4;
HCO2 = 5.15e3;
DCO2 = 3.53e-9;
DHCO3 = 1.09e-9;
DCO32 = 9.6e-8;
KSPCaSO3 = 1.07e-7;
BETCaSO3 = 10;
b(3) = 8.825e-3;
BETCaCO3 = 12.54;
MWCaCO3 = 100.0869;
b(4) = 0.84;
KCO2 = 1.7e-3;
KHCO3 = 6.55e-8;
KSO2 = 6.24;
KHSO3 = 5.68e-5;
Kw = 5.3e-8;
F = @(t, Y, YP) f(t, Y, YP, V_Headspace, F_rate, CSO2_in, CCO2_in, CCa2, R, T, HSO2, b(1), DCa2, DSO2, DHSO3, DSO32, b(2), kLa_CO2, HCO2, DCO2, DHSO3, DSO32, KSPCaSO3, BETCaSO3, b(3), BETCaCO3, MWCaCO3, b(4), KCO2, KHCO3, KSO2, KHSO3, Kw);
vars;
%b0est = [0; 0; 3.3351e-1; 4.879e-6; 4.879e-6; 4.862e+1; 0; 7.413e-6];
b0est = [0;0;1;1;1;1;0;1;];
bp0est = zeros(8,1);
opt = odeset('RelTol', 10.0^(-7), 'AbsTol' , 10.0^(-7));
[b0, bp0] = decic(F, 0, b(5:12), [], bp0est, [], opt);
[tSol,bSol] = ode15i(F, ExperimentalTime, b0, bp0, opt);
for k = 1:origVars
S{k} = char(vars(k));
end
eqns0 = [4.14e-6; 8.4e-4; 8.825e-3; 0.84; 0; 0; 3.3351e-1; 4.879e-6; 4.879e-6; 4.862e+1; 0; 7.413e-6];
B = lsqcurvefit(F, eqns0, ExperimentalTime, ExperimentalConcentrations, zeros(size(eqns0)), inf(size(eqns0)))
%B = lsqcurvefit(F, eqns0, ExperimentalTime, ExperimentalConcentrations)
fprintf(1, '\n\tParameters:\n\t\tkga = %11.3E\n\t\tkLa = %11.3E\n\t\tktot = %11.3E\n\t\tKad = %11.3E\n\tInitial Conditions:\n\t\tCSO2gas = %11.3E\n\t\tCCO2gas = %11.3E\n\t\tStotal = %11.3E\n\t\tCtotal = %11.3E\n\t\tCatotal = %11.3E\n\t\tCCaCO3 = %11.3E\n\t\tCCaSO3 = %11.3E\n\t\tCH = %11.3E\n', B);
F = @(t, B, BP) f(t, B, BP, V_Headspace, F_rate, CSO2_in, CCO2_in, CCa2, R, T, HSO2, DCa2, DSO2, DHSO3, DSO32, kLa_CO2, HCO2, DCO2, DHSO3, DSO32, KSPCaSO3, BETCaSO3, BETCaCO3, MWCaCO3, KCO2, KHCO3, KSO2, KHSO3, Kw);
[tSol,bSol] = ode15i(F, ExperimentalTime, B(5:12), bp0, opt);
figure(1)
plot(ExperimentalTime, ExperimentalConcentrations, 'p')
hold on
plot(tSol, bSol, '--')
hold off
grid
legend('Exp-SO_{2-gas}','Exp-CO_{2-gas}','Exp-S_{total}','Exp-C_{total}','Exp-Ca^{2+}_{total}','Exp-CaCO_{3}','Exp-CaSO_{3}.^{1}/_{2}H_{2}O','Exp-H^{+}', 'Mod-SO_{2-gas}','Mod-CO_{2-gas}','Mod-S_{total}','Mod-C_{total}','Mod-Ca^{2+}_{total}','Mod-CaCO_{3}','Mod-CaSO_{3}.^{1}/_{2}H_{2}O','Mod-H^{+}', 'Location','best')
end
I can not spot where I am going wrong. I do not know where to fix. What can I try?
When I run the code without including the lsqcurvefit, the code does not give errors, however, the experimental results does not match the calculated values. I think that when I regress 4 input values, the calculated values will fit the experimental values. Below is the code on which I have commented-out the lsqcurvefit:
%function EstimationOfParametersAndInitialConditionsUsingLSQCURVEFIT2
function ode15ifun
ExperimentalTime = [600
1200
1800
2400
3000
10200
17400
24600
31800
39000
46200
53400
60600
67800
75000
82200
89400
96600
103800
111000
118200
125400
132600
139800
147000
154200
161400
168600
175800
183000];
ExperimentalConcentrations = [ 0.00E+00 6.97E-08 3.34E-01 4.88E-06 4.88E-06 4.86E+01 0.00E+00 7.41E-06
1.72E-02 8.76E-08 6.79E-01 4.88E-06 4.88E-06 4.92E+01 0.00E+00 9.33E-06
1.71E-02 1.18E-07 1.04E+00 5.11E-06 5.11E-06 4.94E+01 0.00E+00 1.20E-05
1.70E-02 1.03E-07 1.41E+00 5.11E-06 5.11E-06 4.96E+01 0.00E+00 1.05E-05
1.68E-02 1.77E-07 6.00E+00 5.35E-06 5.35E-06 3.78E+01 6.30E+00 1.74E-05
1.74E-02 3.65E-07 1.07E+01 5.35E-06 5.35E-06 3.09E+01 9.33E+00 3.72E-05
1.65E-02 3.50E-07 1.56E+01 5.36E-06 5.36E-06 2.67E+01 1.13E+01 3.55E-05
1.67E-02 8.83E-07 2.07E+01 5.36E-06 5.36E-06 1.98E+01 1.43E+01 1.00E-04
1.67E-02 5.01E-06 2.59E+01 5.07E-06 5.07E-06 1.14E+01 1.79E+01 4.07E-02
1.70E-02 5.06E-06 3.14E+01 5.07E-06 5.07E-06 9.58E+00 1.91E+01 2.45E-01
1.80E-02 4.93E-06 3.67E+01 4.93E-06 4.93E-06 5.65E+00 2.09E+01 6.17E-01
2.15E-02 5.29E-06 4.18E+01 5.30E-06 5.30E-06 3.32E+00 2.21E+01 1.32E+00
2.68E-02 5.30E-06 4.66E+01 5.30E-06 5.30E-06 0.00E+00 2.37E+01 2.29E+00
3.07E-02 5.30E-06 5.09E+01 5.30E-06 5.30E-06 0.00E+00 2.40E+01 2.34E+00
3.33E-02 9.45E-06 5.51E+01 9.46E-06 9.46E-06 0.00E+00 2.43E+01 2.40E+00
3.68E-02 9.45E-06 5.90E+01 9.46E-06 9.46E-06 0.00E+00 2.47E+01 1.82E+00
4.14E-02 1.13E-05 6.23E+01 1.13E-05 1.13E-05 0.00E+00 2.50E+01 1.38E+00
4.48E-02 1.15E-05 6.56E+01 1.15E-05 1.15E-05 0.00E+00 2.54E+01 2.09E+00
4.87E-02 1.10E-05 6.87E+01 1.10E-05 1.10E-05 0.00E+00 2.58E+01 1.82E+00
5.19E-02 1.10E-05 7.12E+01 1.10E-05 1.10E-05 0.00E+00 2.62E+01 1.58E+00
5.47E-02 1.04E-05 7.36E+01 1.04E-05 1.04E-05 0.00E+00 2.67E+01 2.29E+00
5.75E-02 1.04E-05 7.59E+01 1.04E-05 1.04E-05 0.00E+00 2.71E+01 1.62E+00
5.97E-02 1.04E-05 7.78E+01 1.04E-05 1.04E-05 0.00E+00 2.76E+01 1.12E+00
6.17E-02 1.03E-05 7.95E+01 1.03E-05 1.03E-05 0.00E+00 2.81E+01 8.91E-01
6.33E-02 1.04E-05 8.11E+01 1.04E-05 1.04E-05 0.00E+00 2.86E+01 8.51E-01
6.44E-02 1.05E-05 8.24E+01 1.06E-05 1.06E-05 0.00E+00 2.90E+01 7.59E-01
6.48E-02 1.13E-05 8.24E+01 1.13E-05 1.13E-05 0.00E+00 2.90E+01 8.71E-01
6.53E-02 1.13E-05 8.23E+01 1.13E-05 1.13E-05 0.00E+00 2.90E+01 1.12E+00
6.57E-02 1.13E-05 8.21E+01 1.13E-05 1.13E-05 0.00E+00 2.90E+01 1.00E+00
6.59E-02 1.03E-05 8.20E+01 1.03E-05 1.03E-05 0.00E+00 2.90E+01 8.51E-01];
%-------------------------------------------------------------------------------------------------------------------------------------
syms CSO2_gas(t) CCO2_gas(t) S_total(t) C_total(t) Ca2_total(t) CCaCO3(t) CCaSO3(t) CH(t) V_Headspace F_rate CSO2_in CCO2_in R T HSO2 kga DCa2 DSO2 DHSO3 DSO32 kLa_SO2 kLa_CO2 HCO2 DCO2 DHCO3 DCO32 KSPCaSO3 BETCaSO3 ktot BETCaCO3 MWCaCO3 Kad KCO2 KHCO3 KSO2 KHSO3 Kw CCa2
eqn1 = diff(CSO2_gas(t),t)== (F_rate/ V_Headspace) * ( CSO2_in - CSO2_gas(t)) - (( CSO2_gas(t) * R * T - HSO2 * ((S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))) / ((1/kga) + 1/( kLa_SO2 * ((( DSO2 * (HSO2 * CSO2_gas(t) * R * T - ((S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))) + DHSO3 * (((KSO2 * HSO2 * CSO2_gas(t) * R * T/CH(t)) - ((S_total(t) * KSO2 * CH(t))/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)))) + DSO32 * (((( KSO2 * KHSO3 * HSO2 * CSO2_gas(t) * R * T/CH(t)^2) - ((S_total(t) * KSO2 * KHSO3)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)))))) / ( DSO2 * ( HSO2 * CSO2_gas(t) * R * T - ((S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)))))))));
eqn2 = diff(CCO2_gas(t) ,t)== (F_rate/ V_Headspace) * ( CCO2_in - CCO2_gas(t)) - ( kLa_CO2 * ((((DCO2 * (HCO2 * CCO2_gas(t) * R * T - ((C_total(t) * CH(t)^2)/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3))) + DHCO3 * (((KCO2 * HCO2 * CCO2_gas(t) * R * T/CH(t)) - ((C_total (t) * KCO2 * CH(t))/(CH(t)^2 + KCO2 * CH(t) + KCO2*KHCO3)))) + DCO32 * ((((KCO2 * KHCO3 * HCO2 * CCO2_gas(t) * R * T/CH(t)^2) - ((C_total(t) * KCO2 * KHCO3)/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3)))))) / (DCO2 * (HCO2 * CCO2_gas(t) * R * T - ((C_total (t) * CH(t)^2)/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3))))))) * (CCO2_gas(t) * R * T / HCO2 - (( C_total(t) * CH(t)^2)/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3))));
eqn3 = diff(S_total(t),t)== (( CSO2_gas(t) * R * T - HSO2 * ((S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))) / ((1/kga) + 1/(kLa_SO2 * ((( DSO2 * (HSO2 * CSO2_gas(t) * R * T - ((S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))) + DHSO3 * ((( KSO2 * HSO2 * CSO2_gas(t) * R * T/CH(t)) - ((S_total(t) * KSO2 * CH(t))/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)))) + DSO32 * (((( KSO2 * KHSO3 * HSO2 * CSO2_gas(t) * R * T/CH(t)^2) - ((S_total(t) * KSO2 * KHSO3)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)))))) / ( DSO2 * ( HSO2 * CSO2_gas(t) * R * T - (( S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)))))))) - ( 0.162 * exp(-5153 / T) * ((( CCa2 * ((S_total(t) * KSO2 * KHSO3)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))) / KSPCaSO3) - 1)^2 * (BETCaSO3/((CCa2 * ((((( KSO2 * KHSO3 * HSO2 * CSO2_gas(t) * R * T/CH(t)^2) - ((S_total(t) * KSO2 * KHSO3)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)))))) / (DSO2 * (HSO2 * CSO2_gas(t) * R * T - ((S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))))))))/ KSPCaSO3);
eqn4 = diff(C_total(t),t)== ( kLa_CO2 * ((((DCO2 * ( HCO2 * CCO2_gas(t) * R * T - ((C_total(t) * CH(t)^2) / (CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3))) + DHCO3 * (((KCO2 * HCO2 * CCO2_gas(t) * R * T / CH(t)) - ((C_total (t) * KCO2 * CH(t))/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3)))) + DCO32 * ((((KCO2 * KHCO3 * HCO2 * CCO2_gas(t) * R * T / CH(t)^2) - (( C_total(t) * KCO2 * KHCO3)/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3)))))) / ( DCO2 * (HCO2 * CCO2_gas(t) * R * T - ((C_total (t) * CH(t)^2)/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3))))))) * (CCO2_gas(t) * R * T / HCO2 - ((C_total(t) * CH(t)^2)/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3)))) + (ktot * BETCaCO3 * MWCaCO3 * CCaCO3(t) * CH(t) *(1 - (Kad * CH(t))/(1 + Kad * CH(t))));
eqn5 = diff(Ca2_total(t),t)== (-1) * ( ktot * BETCaCO3 * MWCaCO3 * CCaCO3(t) * CH(t) * (1 - (Kad * CH(t))/(1 + Kad * CH(t)))) - (0.162 * exp(-5153 / T) * ((( CCa2 * ((S_total(t) * KSO2 * KHSO3)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))) / KSPCaSO3) - 1)^2 * (BETCaSO3/((CCa2 * (((((KSO2 * KHSO3 * HSO2 * CSO2_gas(t) * R * T/CH(t)^2) - ((S_total(t) * KSO2 * KHSO3)/(CH(t)^2 + KSO2 * CH(t) + KSO2*KHSO3)))))) / (DSO2 * (HSO2 * CSO2_gas(t) * R * T - ((S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))))))))/ KSPCaSO3;
eqn6 = diff(CCaCO3(t),t)== (-1) * (ktot * BETCaCO3 * MWCaCO3 * CCaCO3(t) * CH(t) * (1 - (Kad * CH(t)) / (1 + Kad * CH(t))));
eqn7 = diff(CCaSO3(t),t)== 0.162 * exp(-5153 / T) * ((((CCa2 * ((S_total(t) * KSO2 * KHSO3)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))) / KSPCaSO3) - 1)^2 * (BETCaSO3 / ((CCa2 * (((((KSO2 * KHSO3 * HSO2 * CSO2_gas(t) * R * T/CH(t)^2) - ((S_total(t) * KSO2 * KHSO3) / (CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)))))) / (DSO2 * (HSO2 * CSO2_gas(t) * R * T - ((S_total(t) * CH(t)^2)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3))))))))/ KSPCaSO3;
eqn8 = diff(CH(t),t)== CH(t) + 2 * CCa2 - ((S_total(t) * KSO2 * CH(t)) / (CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)) - 2 * ((S_total(t) * KSO2 * KHSO3)/(CH(t)^2 + KSO2 * CH(t) + KSO2 * KHSO3)) - ((C_total(t) * KCO2 * CH(t))/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3)) - 2 * ((C_total(t) * KCO2 * KHCO3)/(CH(t)^2 + KCO2 * CH(t) + KCO2 * KHCO3))- Kw / CH(t);
eqns = [eqn1 eqn2 eqn3 eqn4 eqn5 eqn6 eqn7 eqn8];
vars = [CSO2_gas(t); CCO2_gas(t); S_total(t); C_total(t); Ca2_total(t); CCaCO3(t); CCaSO3(t); CH(t)];
origVars = length(vars);
M = incidenceMatrix(eqns, vars);
[eqns, vars] = reduceDifferentialOrder(eqns, vars);
isLowIndexDAE(eqns,vars);
f = daeFunction(eqns,vars, V_Headspace, F_rate, CSO2_in, CCO2_in, CCa2, R, T, HSO2, kga, DCa2, DSO2, DHSO3, DSO32, kLa_SO2, kLa_CO2, HCO2, DCO2, DHCO3, DCO32, KSPCaSO3, BETCaSO3, ktot, BETCaCO3, MWCaCO3, Kad, KCO2, KHCO3, KSO2, KHSO3, Kw);
V_Headspace = 1.5e-6;
F_rate = 1.66667e-5;
CSO2_in = 6.51332e-2;
CCO2_in = 0;
CCa2 = 10;
R = 8.314;
T = 323.15;
HSO2 = 149;
kga = 4.14e-6;
DCa2 = 1.39e-9;
DSO2 = 2.89e-9;
DHSO3 = 2.89e-9;
DSO32 = 2.89e-9;
kLa_SO2 = 8.4e-4;
kLa_CO2 = 9.598e-4;
HCO2 = 5.15e3;
DCO2 = 3.53e-9;
DHCO3 = 2.89e-9;
DCO32 = 2.89e-9;
KSPCaSO3 = 1.07e-7;
BETCaSO3 = 10;
ktot = 8.825e-3;
BETCaCO3 = 12.54;
MWCaCO3 = 100.0869;
Kad = 0.84;
KCO2 = 1.7e-3;
KHCO3 = 6.55e-8;
KSO2 = 6.24;
KHSO3 = 5.68e-5;
Kw = 5.3e-8;
F = @(t, Y, YP) f(t, Y, YP, V_Headspace, F_rate, CSO2_in, CCO2_in, CCa2, R, T, HSO2, kga, DCa2, DSO2, DHSO3, DSO32, kLa_SO2, kLa_CO2, HCO2, DCO2, DHSO3, DSO32, KSPCaSO3, BETCaSO3, ktot, BETCaCO3, MWCaCO3, Kad, KCO2, KHCO3, KSO2, KHSO3, Kw);
vars;
y0est = [0; 0; 3.3351e-1; 4.879e-6; 4.879e-6; 4.862e+1; 0; 7.413e-6];
yp0est = zeros(8,1);
opt = odeset('RelTol', 10.0^(-7), 'AbsTol' , 10.0^(-7));
[y0, yp0] = decic(F, 0, y0est, [], yp0est, [], opt);
[tSol,ySol] = ode15i(F, [600, 183000], y0, yp0, opt);
for k = 1:origVars
S{k} = char(vars(k));
end
figure(1)
plot(tSol,ySol)
hold on
plot(ExperimentalTime, ExperimentalConcentrations, 'p')
legend('Exp-SO_{2-gas}','Exp-CO_{2-gas}','Exp-S_{total}','Exp-C_{total}','Exp-Ca^{2+}_{total}','Exp-CaCO_{3}','Exp-CaSO_{3}.^{1}/_{2}H_{2}O','Exp-H^{+}', 'Mod-SO_{2-gas}','Mod-CO_{2-gas}','Mod-S_{total}','Mod-C_{total}','Mod-Ca^{2+}_{total}','Mod-CaCO_{3}','Mod-CaSO_{3}.^{1}/_{2}H_{2}O','Mod-H^{+}', 'Location','best')
hold off
end
%eqns0 = [4.14e-6; 8.4e-4; 8.825e-3; 0.84; 0; 0; 3.3351e-1; 4.879e-6; 4.879e-6; 4.862e+1; 0; 7.413e-6];
%B = lsqcurvefit(@ode15ifun, eqns0, ExperimentalTime, ExperimentalConcentrations, zeros(size(eqns0)), inf(size(eqns0)))
%F = @(t, Y, YP) f(t, B, BP, V_Headspace, F_rate, CSO2_in, CCO2_in, CCa2, R, T, HSO2, B(1), DCa2, DSO2, DHSO3, DSO32, B(2), kLa_CO2, HCO2, DCO2, DHSO3, DSO32, KSPCaSO3, BETCaSO3, B(3), BETCaCO3, MWCaCO3, B(4), KCO2, KHCO3, KSO2, KHSO3, Kw);
%[tSol,eqns] = ode15i(F, ExperimentalTime, B(5:12), bp0, opt);
%figure(2)
%plot(ExperimentalTime, ExperimentalConcentrations, 'p')
%hold on
%plot(tSol, ySol, '--')
%hold off
%grid
%end

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