Ch E 310 - Fall 10 - Lecture 5

Ch E 310 - Fall 10 - Lecture 5 - Lecture 5 September 7,...

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Lecture 5 – September 7, 2010 Agenda: Solving linear systems of equations: A·x = b Example: mass balance on a process flow Inverting matrices Calculating determinants Cramer’s Rule In-Class Exercise 4 Remember that HW 1 is due at noon on WebCT
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Review: Matrix Multiplication Matrix math is especially important in solving linear systems Most linear systems will take the matrix equation form A x = b First equation: a 11 x 1 + a 12 x 2 + a 13 x 3 + a 14 x 4 + a 15 x 5 = b 1 Second equation: a 21 x 1 + a 22 x 2 + a 23 x 3 + a 24 x 4 + a 25 x 5 = b 2 Matrix notation is a more compact and efficient way of writing (and solving) systems of linear equations 5 4 3 2 1 5 4 3 2 1 55 54 53 52 51 45 44 43 42 41 35 34 33 32 31 25 24 23 22 21 15 14 13 12 11 b b b b b x x x x x a a a a a a a a a a a a a a a a a a a a a a a a a b x A
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Solving Linear Systems of Equations c 3 c 1 c 5 c 2 c 4 q 01 = 7 c 01 = 10 q 03 = 8 c 03 = 20 q 5 = 2 q 4 = 13 q 24 = 3 q 54 = 2 q 34 = 8 q 23 = 1 q 25 = 1 q 12 = 5 q 15 = 3 q 31 = 1 Example: System of well-mixed process units (CSTRs) Given concentrations ( c ), flows ( q ): solve for c i in the units
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Solving Linear Systems of Equations Write down system of equations based on flow diagram Material balances around each process unit ( in = out ): Unit 1: q 01 c 01 + q 31 c 3 q 15 c 1 q 12 c 1 = 0 Unit 2: q 12 c 1 q 25 c 2 q 24 c 2 q 23 c 2 = 0 Unit 3: q 03 c 03 + q 23 c 2 q 31 c 3 q 34 c 3 = 0 Unit 4: q 24 c 2 + q 34 c 3 + q 54 c 5 q 4 c 4 = 0 Unit 5: q 15 c 1 + q 25 c 2 q 54 c 5 q 5 c 5 = 0 5 equations and 5 unknowns since all flows ( q i ) are given With the equations written down, how should we solve this? The equations cannot be solved by simple substitution
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Ch E 310 - Fall 10 - Lecture 5 - Lecture 5 September 7,...

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