1 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. CHAPTER 12 12.1 Flow balances can be used to determine Q01 = 6 Q15 = 3 Q12 = 4 Q31 = 1 Q03 = 8 Q25 = 1 Q23 = 1 Q54 = 2 Q55 = 2 Q24 = 2 Q34 = 8 Q44 = 12 Mass balances can be used to determine the following simultaneous equations, 1234570 100 12044000 001900 48028122 031004 0ccccc The solution and the matrix inverse can then be developed. For example, using MATLAB, >> A=[7 0 -1 0 0;-4 4 0 0 0;0 -1 9 0 0;0 -2 -8 12 -2;-3 -1 0 0 4]; >> B=[120;0;48;0;0]; >> C=A\B C = 18.1935 18.1935 7.3548 10.9677 18.1935 >> inv(A) ans = 0.1452 0.0040 0.0161 0 0 0.1452 0.2540 0.0161 0 0 0.0161 0.0282 0.1129 0 0 0.0591 0.0722 0.0806 0.0833 0.0417 0.1452 0.0665 0.0161 0 0.2500 12.2 The relevant coefficients of the matrix inverse are a23–1 = 0.01887 and a43–1 = 0.08748. Therefore, a 25% change in the input to reactor 3 will lead to the following concentration changes to reactors 2 and 4: 20.01887(0.25 160) 0.754717c 40.08748(0.25 160) 3.499142c These can be expressed as percent changes, 220.754717100% 100% 6.56%11.50943cc 443.499142100% 100% 20.59%16.99828cc 12.3 Because of conservation of flow: 01 03 44 55QQ QQ 12.4 Mass balances can be used to determine the following simultaneous equations,2 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 1234580 300 5044 0 0 0 0021000 16000 7103 042006 0ccccc The solution can then be developed. For example, using MATLAB, >> A=[8 0 -3 0 0;-4 4 0 0 0;0 -2 10 0 0;0 0 -7 10 -3;-4 -2 0 0 6]; >> B=[50;0;160;0;0]; >> C=A\B C = 13.2432 13.2432 18.6486 17.0270 13.2432 12.5 Flow balances can be used to determine Q01 = 5 Q15 = 3 Q12 = 0 Q31 = 2 Q03 = 8 Q25 = 0 Q23 = 7 Q54 = 0 Q55 = 3 Q24 = 7 Q34 = 3 Q44 = 10 Mass balances can be used to determine the following simultaneous equations, 1234550000 5007 700 02 0 10 0 0 160073100 030 0 03 0ccccc The solution can then be developed. For example, using MATLAB, >> A=[5 0 0 0 0;0 7 -7 0 -1;-2 0 10 0 0;0 -7 -3 10 0;-3 0 0 0 3]; >> B=[50;0;160;0;0]; >> C=A\B C = 10.0000 18.0000 18.0000 18.0000 10.0000 12.6 Mass balances can be written for each of the reactors as 13 1 12 1 21 2400 0Qc Qc Qc 12 1 21 2 23 20Qc Qc Qc 13 1 23 2 33 3200 0Qc Qc Qc Values for the flows can be substituted and the system of equations can be written in matrix form as3 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. 123120 20 0 40080 80 0 040 60 120 200ccc The solution can then be developed. For example, using MATLAB, clc A=[120 -20 0;-80 80 0;-40 -60 120]; B=[400;0;200]; C=A\B C = 4.0000 4.0000 5.0000 12.7 Mass balances can be written for each of the lakes as Superior, c1: 1180 67c Michigan, c2: 2710 36c Huron, c3: 12 3740 67 36 161cc c Erie, c4: 343850 161 182cc Ontario, c5: 454720 182 212cc The system of equations can be written in matrix form as 1234567 0 0 0 0 180036 0 0 0 71067 36 161 0 0 7400 0 161 182 0 38500 0 0 182 212 4720ccccc The solution can then be developed. For example, using MATLAB, >> A=[67 0 0 0 0; 0 36 0 0 0; -67 -36 161 0 0; 0 0 -161 182 0; 0 0 0 -182 212]; >> B=[180 710 740 3850 4720]'; >> C=A\B C = 2.6866 19.7222 10.1242 30.1099 48.1132 12.8 (a) The solution can be developed using your own software or a package. For example, using MATLAB, >> A=[13.422 0 0 0; -13.422 12.252 0 0; 0 -12.252 12.377 0; 0 0 -12.377 11.797]; >> W=[750.5 300 102 30]';4 PROPRIETARY MATERIAL. © The McGraw-Hill Companies, Inc. All rights reserved. No part of this Manual may be displayed, reproduced or distributed in any form or by any means, without the prior written permission of the publisher, or used beyond the limited distribution to teachers and educators permitted by McGraw-Hill for their individual course preparation. If you are a student using this Manual, you are using it without permission. >> AI=inv(A) AI = 0.0745 0 0 0 0.0816 0.0816 0 0 0.0808 0.0808 0.0808 0 0.0848 0.0848 0.0848 0.0848 >> C=AI*W C = 55.9157 85.7411 93.1163 100.2373 (b) The element of the matrix that relates the concentration of Havasu (lake 4) to the loading of Powell (lake 1) is a41–1 = 0.084767. This value can be used to compute how much the loading to Lake Powell must be reduced
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