|
Concentrations and chemical potentials during
glass melting
Problem
A certain number of chemical substances are mixed with given
concentrations. In the mixture chemical reactions take place and
additional compounds are created depending on the temperature.
We want to calculate the concentrations of all appearing substances
in the melange. From these it is then easy to determine the
chemical potentials, which are necessary for qualitative statements.
Note that the resulting system of equations is badly conditioned and
constrained (concentrations are non-negative).
Application
During glass melting it is possible to predict properties of the
resulting glass after melting and solidification when the
concentrations and chemical potentials of the involved substances are
known. So it is interesting to determine concentrations
of the initial substances such that the glass has "good" properties
after melting and solidification (e.g. good reflection and polishing
behaviour).
Example
We consider an example with 4 initial substances and 16 chemically
possible combinations. Following the model of Prof. R. Conradt (RWTH
Aachen, Cooperation in the framework of a DFG-project) the resulting
non-linear algebraic system of equations consists of 21 equations in
21 indeterminats.
| Initial substances |
C, M, A, S |
| Chemical compounds |
CS, C2S, A3S2, C3A,
C12A7, CA2, CA6, CAS2,
C2AS, M2A2S5, CMS2,
MS, M2S, MA, C2MS2, CMS |
| Point balance | Mol. |
Balance equation
C+M+A+S+CS+C2S+A3S2+C3A+C12A7+CA2+CA6+CAS2+C2AS+M2A2S5+CMS2+MS+M2S+MA+C2MS2+CMS-Mol = 0
Partial balance equations
Let zC, zM, zA, zS be the given concentrations of the initial substances.
C+CS+2*C2S+3*C3A+12*C12A7+CA2+CA6+CAS2+2*C2AS+CMS2+2*C2MS2+CMS-zC = 0
M+2*M2A2S5+CMS2+MS+2*M2S+MA+C2MS2+CMS-zM = 0
A+3*A3S2+C3A+7*C12A7+2*CA2+6*CA6+CAS2+C2AS+2*M2A2S5+MA-zA = 0
S+CS+C2S+2*A3S2+2*CAS2+C2AS+5*M2A2S5+2*CMS2+MS+M2S+2*C2MS2+CMS-zS = 0
Equilibrium equations
3052.274545677636801038644118353301*C*S-CS*Mol = 0
22816621.137207969245212762945173950929*C2*S-C2S*Mol2 = 0
3.257822373244675193654532953387*A3*S2-A3S2*Mol4 = 0
16974.79444127609180291295720843558*C3*A-C3A*Mol3 = 0
118781134623051252286.224424907057709325*C12*A7-C12A7*Mol18 = 0
843.622667036838337524627427318019*C*A2-CA2*Mol2 = 0
936.021558357555341922011682037709*C*A6-CA6*Mol6 = 0
297825.053822998028828276585968099771*C*A*S2-CAS2*Mol3 = 0
104744891.214541505489320983776109122095*C2*A*S-C2AS*Mol3 = 0
419677.171597410943183594686461040144*M2*A2*S5-M2A2S5*Mol8 = 0
11843335.652624021727690155608746433145*C*M*S2-CMS2*Mol3 = 0
202.167460260932835817098052526253*M*S-MS*Mol = 0
95299.115966058548116433240683887213*M2*S-M2S*Mol2 = 0
1.818638731995420206373348982492*M*A-MA*Mol = 0
457378802509.71344211060898513666930629*C2*M*S2-C2MS2*Mol4 = 0
24353698.418928372736312809950149628509*C*M*S-CMS*Mol2 = 0
The computation in
SINGULAR
|