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

Scenario

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

Scenario

Variables that might be of interest:

Temperature

Level

Pressure

Flow

Valve status

v4

v2 v3

v6

v1

v5

T1

L1

P1

F1

v1

Scenario

Variables that might be of interest:

Temperature

Level

Pressure

Flow

Valve status

P1

P3P2

T1

T2

T3

T4

L1

L2

v4

v2 v3

v1

v5

F3F2

F1

F4

F5

T1

L1

P1

F1

v1

v6

L1

L2

v4

v2 v3

v1

v5

F3F2

F1

F4

F5

v6

What kind of stable and direct relationships can we identify?

Flow type relationships:Opening v2 increases F2Opening v3 increases F3Opening v4 increases F4

Level relationships:Increasing F2 decreases L1Increasing F3 decreases L1

Question: Why or why notOpening 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.

Incidence Diagrams• Diagramatic and analytic tool to

describe direct, monotonic, and causal relationships between variables.

x

X

y

y

X

y

x y

“Increasing x causes y to increase”

“Increasing x causes y to decrease”

L1

L2

v4

v2 v3 F3F2

F1

F4

F5

Flow type relationships:Opening v2 increases F2Opening v3 increases F3Opening v4 increases F4

Level relationships:Increasing F2 decreases L1Increasing F3 decreases L1

v2 F2

v3

v4 F4

L1F3

If gravity fed, may have a feedback effect on L1 on the flow..Increasing L1 increases F2Increasing L1 increases F3

T1

Control system may regulate the column temperature with reflux..T1 opens (increases) v2Increasing F2 decreases T1

T1

Control actionPhysical response

L1

L2

v4

v2 v3 F3F2

F1

F4

F5

T1

v2 F2

v3

v4 F4

L1F3

T1

Interpreting an incidence diagram• Break a system into independent or semi-independent parts• Detect consistent chains of effectse.g. if I could force L1 to increase, what would be the effect?

L1

L2

v4

v2 v3 F3F2

F1

F4

F5

T1

v2 F2

v3

v4 F4

L1F3

T1

e.g. if I could force L1 to increase, what would be the effect?

Identify all paths from and to the variable

L1

L2

v4

v2 v3 F3F2

F1

F4

F5

T1

v2 F2

v3

v4 F4

L1F3

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.

Focal adhesion signal transduction pathay in human cellsfromhttp://www.wikipathways.org/images/5/54/Hs_Focal_adhesion_KEGG.svg

A second ChemE exampleIncidence diagram showing the interaction of you reactor conditions on profitability.

Based on this graph alone, changing what variables will consistently increase profitability?

Options:

Purification efficiency -> profitability {+,+}

Consistent effect on profit:• Purification efficiency

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

Consistent effecton 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

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.

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

a b c d e f a f

a b c d e f

ghi

a f

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