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©GoldSim Technology Group LLC., 2012 Network Flows and Linear Programming The Mathematical Madness behind the Magic GoldSim Technology Group

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©GoldSim Technology Group LLC., 2012 Objective of the Flow Module Given a system of discrete locations connected by conduits of flowing material… …determine the “optimal” flow of material through that network.

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©GoldSim Technology Group LLC., 2012

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Flow SolverGoldSim 10.5: Solve using iteration

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©GoldSim Technology Group LLC., 2012 Benefits Optimal allocation of material Mass conservation Integrates handling of flow and transport Built-in storage functions Integrated handling of priorities and costs Influence lines represent flows

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©GoldSim Technology Group LLC., 2012 What do we mean by “Optimal”? Meaning 1: Maximal Profit (e.g., commodity distribution) –If the network is controlled by a single operator selling to multiple customers, then the goal is to maximize profit. –Example: Natural gas distributor (PSE) Meaning 2: Prioritized Flow (e.g., water distribution) –In this case water is divided up based on various users’ priorities: Priority 1 users get first dibs on water until all their demands are met… …and so on until the lowest priority (farmers) get what’s left over. The prioritized flow method uses the same underlying functions as maximal profit

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©GoldSim Technology Group LLC., 2012 Flowing “Media” The quantity of liquids and/or solids that flow from one discrete location to another. Examples: water, CO 2, rocks, sediment in water. Assume incompressible and volume is additive, taking porosity of any solid media into account. –For example, 1 gallon of water dumped into a tank containing 1 gallon of sediment whose porosity is 0.3 would consume a total volume of 1.7 gallons (1 gal of Water + (1 – 0.3)*1 gal of Sediment).

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©GoldSim Technology Group LLC., 2012 Cells (Any Flow Network Elements) Model elements that produce, consume, store, or route fluid. Examples: –Pump –Evaporation –Detention pond –A city –Stockpile

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©GoldSim Technology Group LLC., 2012 Flows (Influence Lines)

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©GoldSim Technology Group LLC., 2012 Flow Capacity and Costs

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©GoldSim Technology Group LLC., 2012 Source Cells A source cell feeds fluid into the system. Source cells have infinite supply, but their outflow rate(s) may be limited. Examples: –Rainfall in a particular geographic area –CO2 from a power plant –Sediments from erosion

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©GoldSim Technology Group LLC., 2012 Sink Cells A sink cell removes fluid from the system. The capacity of sink to absorb fluid is infinite, but the inflow rate may be limited. Examples: –Evaporation –Outflow from a river (model boundary) –Consumers

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©GoldSim Technology Group LLC., 2012 Zero-Volume Cells (Routers)

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©GoldSim Technology Group LLC., 2012 User-Specified Cells Cells found in current GoldSim version (CT module) Distinct from Routers, which have zero volume Implications on CT models (need volume for concentration to make sense)

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©GoldSim Technology Group LLC., 2012 Dynamic Volume Cells (Stores)

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©GoldSim Technology Group LLC., 2012 Store Cells Attributes

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©GoldSim Technology Group LLC., 2012 Demand Priorities and Revenues

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©GoldSim Technology Group LLC., 2012 Simple Example…

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Internal Flow: Mass Transfer Chapter 8 Section 8.9.

Internal Flow: Mass Transfer Chapter 8 Section 8.9.

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