Robotics
Circuit Design
Medicine
Glass Melting
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 balanceMol.

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

Lille, 08-07-02 http://www.singular.uni-kl.de