1. Modeling Basics: 2. Incidence Graphs By Peter Woolf (pwoolf@umich.edu) University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative commons

2. Scenario After creating a detailed verbal description of your process, you want to map out and interpret how the key variables influence each other.

3. Scenario Variables that might be of interest: Temperature Level Pressure Flow Valve status v4 v2v3 v6 v1 v5 T1 L1 P1 F1 v1

4. Scenario Variables that might be of interest: Temperature Level Pressure Flow Valve status P1 P3 P2 T1 T2 T3 T4 L1 L2 v4 v2v3 v1 v5 F3F2 F1 F4 F5 T1 L1 P1 F1 v1 v6

5. L1 L2 v4 v2v3 v1 v5 F3F2 F1 F4 F5 v6 What kind of stable and direct relationships can we identify? Flow type relationships: Opening v2 increases F2 Opening v3 increases F3 Opening v4 increases F4 Level relationships: Increasing F2 decreases L1 Increasing F3 decreases L1 Question: Why or why not Opening v2 decreases L1? Although true, the effect is indirect. The effect on L1 is directly caused by F2 and F3, and indirectly by v2 and v3.

6. Incidence Diagrams Diagramatic and analytic tool to describe direct, monotonic, and causal relationships between variables. x X y y X y xy “Increasing x causes y to increase” “Increasing x causes y to decrease”

7. L1 L2 v4 v2v3 F3F2 F1 F4 F5 Flow type relationships: Opening v2 increases F2 Opening v3 increases F3 Opening v4 increases F4 Level relationships: Increasing F2 decreases L1 Increasing F3 decreases L1 v2F2 v3 v4F4 L1 F3 If gravity fed, may have a feedback effect on L1 on the flow.. Increasing L1 increases F2 Increasing L1 increases F3 T1 Control system may regulate the column temperature with reflux.. T1 opens (increases) v2 Increasing F2 decreases T1 T1 Control action Physical response

8. L1 L2 v4 v2v3 F3F2 F1 F4 F5 T1 v2F2 v3 v4F4 L1 F3 T1 Interpreting an incidence diagram Break a system into independent or semi- independent parts Detect consistent chains of effects e.g. if I could force L1 to increase, what would be the effect?

9. L1 L2 v4 v2v3 F3F2 F1 F4 F5 T1 v2F2 v3 v4F4 L1 F3 T1 e.g. if I could force L1 to increase, what would be the effect? Identify all paths from and to the variable

10. L1 L2 v4 v2v3 F3F2 F1 F4 F5 T1 v2F2 v3 v4F4 L1 F3 T1 e.g. if I could force L1 to increase, what would be the effect? Identify all paths from and to the variable (1) L1-> F2 -| L1 (2) L1 -> F3 -| L1 (3) L1 -> F2 -| T1 ->v2->F2 -|L1 {+, +, -} {+, +, -,-,-,+} Inconsistent path: system will resist change Consistent path: system will reenforce change Side effect: forcing L1 to increase will cool the column causing v2 to close, decreasing F2, and increasing L1.

11. Focal adhesion signal transduction pathay in human cells from http://www.wikipathways.org/images/5/54/Hs_Focal_adhesion_KEGG.svg

12. A second ChemE example Incidence diagram showing the interaction of you reactor conditions on profitability. Based on this graph alone, changing what variables will consistently increase profitability?

13. Options: Purification efficiency -> profitability {+,+}{+,+} Consistent effect on profit: Purification efficiency

14. Consistent effect on profit: Purification efficiency Mixing speed {+,+,+} {+,-,+,+} {+,+,-,+,+} {+,-,-,-} Options: Mixing speed ->purification efficiency -> profitability Mixing speed -| culture density-| purification efficiency -> profitability Mixing speed -> final broth pH -| culture density-| purification efficiency -> profitability Mixing speed -| culture density -> fermentation temp -> mixing speed

15. Consistent effect on profit: Purification efficiency Mixing speed Inconsistent: culture density {+,-,-} {+,+,+,+,+} {+,+,+,-} {+,+,+,+,-} Options: Culture density -| purification efficiency -> profitability culture density-> fermentation temperature -> mixing speed -> purification eff -> profitability culture density-> fermentation temperature -> mixing speed -| culture density culture density-> fermentation temperature -> mixing speed -> pH -| culture density

16. Check the rest! Consistent effect on profit Inconsistent effect on profit Note! This does not mean that these are the only ways to increase the profitability of the plant, but instead these are the only consistent ways to increase profit according to this diagram.

17. Consistent vs. inconsistent networks Whole networks can be consistent or inconsistent. Consistent networks have the advantage that they can be collapsed into a single conceptual unit abcdefafabcdef gh i af

18. From http://controls.engin.umich.edu/wiki/images/6/61/ConsistentGraphs.doc