GES 132 Engineering Problem Solving with MATLAB

Transcription

1 GES 132 Engineering Problem Solving with MATLAB Project III Material Balances: Charge Calculation in Melting Furnaces Nagy El-Kaddah

2 Charge Calculation in Melting Furnaces Induction, Electric-arc and Plasma crucible furnaces are widely used for the batch melting of ferrous and non-ferrous metals. At the beginning of the heat, the furnace is charged with melting stock usually composed of commercially pure primary metal and scrap from one or more sources. Upon melting the metal charge, the alloy composition is adjusted by adding master alloys. The amount to various materials to be charged depends on the furnace capacity, and the composition of the charge materials in the plant. Plasma crucible furnace

3 Mass Conservation Law The calculation of charge materials is based on the principal of mass conservation, which for batch processes states: Mass in = Mass out Since the alloy contains many species (for example steel contains carbon, silicon, etc), we end up with a set of equations for total mass and each species in the alloy. For example, if we have three charging materials, M1, M2, M3, the total mass balance is wt. of M1 + wt. of M2 + wt. of M3 = wt. of produced metal The species {i} balance equation is wt. of M1* i%in M1 + wt. of M2* i%in M2 + wt. of M3* i%in M3 = wt. of produced metal* i%in produced metal

4 Solution Procedure of Mass Balance Problems Draw a box reactor showing the input and output streams of charge materials and products, as shown below. For each input and output stream, write the amount and composition of charge or product materials If the amount of material is not known, give it a name. From this sketch, determine number of unknowns (say n) Develops n balance equations Solve the resultant set of linear equations. Input Materials Input stream 1 amount; analysis Input stream 2 amount; analysis Reactor Output Materials Output stream 1 amount; analysis

5 Application to Melting Furnaces Example: A steel plant produces low carbon steels by melting steel scrap in a 20 tons electric arc furnace. If the following materials are available in the plant. C % Si % Mn% Steel scrap Pig iron Ferrosilicon Ferromanganese calculate the amount of each material to be charged in order to produce steel containing 0.3% C, 0.2%Mn and 0.2%Si. Solution: Input Materials Output Materials Steel Scrap X1 C=0.1 Pig iron X2 C=3.2; Si=1.0; Mn=0.1 Ferrosilicon X3 Si=45 Ferromanganese X4 Mn=60 Electric Arc Furnace Molten Steel 20,000 kg C=0.3; Si=0.2; Mn=0.2

6 Application to Melting Furnaces (cont.) Solution (cont.): # of Unknown 4 The 4 balance equations needed are: Total Balance: X1 + X2 + X3 + X4 = Carbon Balance: 0.1 X X2 + 0 X3 + 0 X4 = 0.3 * Silicon Balance: 0 X X X3 + 0 X4 = 0.2 * Mn Balance: 0 X X2 + 0 X X4= 0.2 * A X = C A = A Simple Matlab program A=[1,1,1,1;.1,3.2,0,0; 0,1.0,45,0;0,.1,0,60]; C=[20000;6000;4000;4000]; X=inv(A)*C; (or X=A\C;) X = X 1 X 2 X 3 X 4 Solution X C = = Scrap Pig iron kg Ferrosilicon Ferromanganese

7 A General Charge Calculation Program Start Enter the analysis of Charge Materials Input amount and composition of steel to be produced Create coefficient and constant Matrics of Balnced equations Solve balance equations Print amounts of charged materials Stop

8 A General Charge Calculation Program(cont.) function charge materials=['steel Scrap '; 'Pig Iron ';'Ferrosilicon ';'Ferromanganese']; scrap_analysis=[.1,0,0]; pig_analysis=[3.2,1,.1]; fsilicon_analysis=[0,45,0]; fmanganese_analysis=[0,0,60]; W= input('enter weight of steel: '); steel_analysis= input('enter chemical composition of steel (C,Si,Mn): '); tmass=[1,1,1,1]; tcarbon=[scrap_analysis(1),pig_analysis(1),fsilicon_analysis(1),fmanganese_analysis(1)]; tsilicon=[scrap_analysis(2),pig_analysis(2),fsilicon_analysis(2),fmanganese_analysis(2)]; tmanganese=[scrap_analysis(3),pig_analysis(3),fsilicon_analysis(3),fmanganese_analysis(3)]; a=[tmass;tcarbon;tsilicon;tmanganese]; C=W*[1;steel_analysis(1);steel_analysis(2);steel_analysis(3)]; X=inv(a)*C; for i=1:4 fprintf('the amount of %s = %8.1f kg \n',materials(i, :),X(i,1)) end

9 A General Charge Calculation Program(cont.) Results for test case Input >> charge Enter weight of steel: Enter chemical composition of steel (C,Si,Mn): [.3,.2,.2] Output The amount of Steel Scrap = kg The amount of Pig Iron = kg The amount of Ferrosilicon = kg The amount of Ferromanganese = kg