MatLab calculations

The formation reactions are the heart of the gasifier. From the formation reactions together with mass balances, a set of equations giving the syngas composition may be derived. To solve such a complex equation set the mathematical computer tool MatLab is used.

Calculating a syngas composition in MatLab demands manually equation derivation. MatLab is only an equation solver and the derivation from formation reaction to a solvable equation set is done manually. The reason for doing this is to show that the results calculated from simple formation reactions corresponds to calculated values from more advanced computer tools as PRO/II and the chemical equilibrium calculator.

3.2.2.1 Complete equation set

From the formation reactions presented in the theoretical background it is possible to calculate the syngas composition. The conversion of coal, oxygen and steam to syngas is decided by these formation reactions together with input values as temperature, pressure and coal composition. The coal composition is assumed to be a combination of C, H, O, N and S. The seven independent formation reactions are presented in 2.2.2.1 Formation reactions and are repeated below.

The reactions contain a total of 12 substances. In theory traces of all 12 substances is found in the syngas composition and the equation set therefore contains a minimum of 12 unknown. In calculations it is not necessary to include all 12 substances in the equilibrium. This is due to neglectable amount of several of the substances. A method containing all 12 substances is although presented before the set of equations is reduced to containing only the relevant ones.

All the reactions have belonging equilibrium equations presented in chapter 2.2.2.2

Equilibrium equations. The equilibrium equations are derived from the formation reactions and the deriving procedure is described in Appendix A. The equilibrium equations are repeated below.

2

The 12 unknown molar fractions are implemented in equation 2.40 to 2.46. Equation 2.40 to 2.46 are also the first 7 equations in the equation set. The equilibrium constants K are tabulated for given temperatures.

The element mass balances for the five feed substances are independent equations and are implemented in the equation set. An element mass balance equation gives that the amount of an element fed to a process must equal the amount of the element going out of the process. If 1kmol/s of C is fed to the gasifier the sum of C, CO, CO2 and CH4 also must equal to

1kmol/s. In this process there are five elements in the feed giving five element mass balances.

C, N and S are in theory fed to the gasifier process only through the coal. H and O are fed both through the coal and through the feeds of oxygen and steam. When calculating the element feeds it is therefore important to include the H and O feed from steam and oxygen in

addition to the feed from the coal. If the calculation should be extremely precise the few percent of nitrogen in the oxygen feed from the ASU and possible pollution in the water also needs to be taken into account.

The equation set now contains 12 equations, but the introduction of the variable giving the total number of moles in the equilibrium gas mixture, n, makes need for an extra equation.

This is solved by the equation of the total mass balance.

2 4 2 2 2 2 2 2 1

C CO CO CH O H H O N NO S SO H S

y +y +y +y +y +y +y +y +y +y +y +y = (3.6)

The equation set now contains 13 equations with 13 unknown.

3.2.2.2 Reduction of the equation set

The equation set containing 13 equations and 13 unknown is very complex to solve. In fact, the extremely small amount of some of the substances makes a MatLab model not able to solve the equation set. When the molar fractions get this close to zero and other mole fractions are divided by these the model is unsolvable. The solution is to reduce the set of equations to a set containing only the substances with practical influence of the syngas composition.

The amount of CH4, NO, H2S and SO2 are considered to small to have major impact on the syngas composition. The remaining independent formation reactions are then 2.30, 2.31 and 2.35.

This assumption also gives that there is no nitrogen or sulfur in the coal composition. This is a simplification need to be done to get the MatLab model to converge. The coal is then

containing of a combination of carbon, hydrogen and oxygen.

It is also assumed that all the oxygen is consumed in the gasification process and there is no oxygen left in the syngas. This reduces the three independent formation reactions to two new formation reactions. These reactions are independent of each other, but are dependent of the previous three.

The first new formation reaction, 2.32, is derived by subtracting 2.31 from 2.30 multiplied with two.

2 2

C CO+ CO (2.32)

The other new formation reaction, 2.24, is derived by subtracting 2.35 from 2.30.

2 2

C H O+ CO H+ (2.34)

It is further assumed that all the carbon is gasified and there is no pure carbon left in the syngas. In real life some of the carbon is not gasified and leaves the gasifier together with ash and slag. It is although a legal approximation assuming no carbon in the syngas.

Subtraction of 2.34 from 2.32 gives a formation reaction without pure carbon.

2 2 2

CO H O+ CO +H (3.10)

From 3.10 the following equilibrium constant can be set up.

2 2

Equation 3.11 contains five unknown. First of all the K value is derived.

[ ]

2

The K value is found by inserting 3.12, 3.13 and 3.14 in 3.11.

2 2 2 2 2 2

This gives an equation with four unknown.

2 2

KCO2, KCO and KH2O are tabulated for specific temperatures. In some literature the equilibrium constant in equation 3.11 are tabulated directly and the derivations performed in 3.12 to 3.16 are not necessary. It is although shown here to give the entire derivation from the elementary formation reaction to the calculated syngas composition.

The next three equations are given by element mass balance and form a new unknown, n.

( 2)

Here is yC,Feed the number of moles of carbon in the coal. yH,Feed is moles of hydrogen in the coal plus the moles of steam multiplied with two. The addition of two times the number of moles of steam is to add on the two hydrogen atoms in H2O. Equally yO,Feed is moles of

oxygen in the coal plus to times the number of moles of O2 plus the number of moles of steam.

The last equation which completes the equation set and is the total mass balance.

2 2 2 1

CO CO H H O

y +y +y +y = (3.20)

The equation set then consists of five equations with five unknown. This is a set of equations solvable in MatLab and a script using the fsolve function is made.

The script with the function to solve the five equations is given in Appendix B. The script which runs the function is given in Appendix C. Results from the calculations are presented and compared with results from other computer tools in the discussion part of the report.