Following the flow of ʻstuffʼ

Transcription

1 MCB 137 MONEY SPRING 2008 Follow the money Following the flow of ʻstuffʼ Most modeling is simply bookkeeping the flows of quantities that can be tracked: mass, number, money, etc. Consider, for example, your savings account. The flow of money (with units $) is diagramed in the FlowChart window shown in Figure 1. Figure 1. the flowchart following the flow of money in and out of a bank account. The three icons required to represent the flow of money are: 1. The Reservoir (units = $) 2. The Flow (units = $/t) 3. The Functions show how the flows and reservoir depend on one another. The units of Functions depend on what icon they modify. 4. Arcs connect functions to flows or reservoirs to show dependences. The Equation Window shows the equations corresponding to the Flowchart: Rate of accumulation: d/dt (Dollars) = +Deposits - Withdrawals + Interest Initial Amount: [$} INIT Dollars = 1000 Inflow: [$/t] Deposits = 2000 Interest = Dollars*Interest_Rate Outflow: [$/t] Withdrawals = Fixed_Expenses + Fees + Spending - 1 -

2 Parameters: [$/t] Interest_Rate = 0.04 Fixed_Expenses = 900 Fees = 10 Spending = Dollars*0.5 Now, suppose you have a foreign bank account. You can transfer funds from your bank account in England, but you have to convert from pounds ( ) to dollars ($). Suppose you transfer 1% of your British assets every month to your American account. Then the flow of money through the two linked bank accounts is shown in Figure 2. Figure 2. The flow of money in two linked bank accounts The Equation Window shows the same information in symbolic form: - 2 / 6-

3 Reservoirs: d/dt (Dollars) = + Deposits - Withdrawls + Interest INIT Dollars = 1000 d/dt (Pounds) = + Pound_Interest Transfers INIT Pounds = 100 Flows: Deposits = Conversion*Transfers+2000 Withdrawls = Fixed_Expenses+Fees+Spending Interest = Dollars*Interest_Rate Pound_Interest = Pound_Interest_Rate*Pounds Transfers = 0.01*Pounds Functions: Fixed_Expenses = 900 Fees = 10 Spending = Dollars*0.5 Interest_Rate = 0.04 Pound_Interest_Rate = 0.08 Conversion = 1.5 Flows of physical quantities The flows of physical quantities are different in a fundamental way from the flow of money in the example above. There the flows were determined by the withdrawal and deposit functions that depend on many external factors, including the whims and psychology of the investor. In most cases, the flow of physical quantities are driven by potentials that can be calculated from physical laws. For example, the flow of heat is driven by temperature differences, the flow of fluids are driven by differences in pressure differences, and the diffusion of particles is driven by differences in concentration. As a consequence of the physical origin of driving forces, you will encounter a common Flowchart structure again and again; it has the general form shown in Figure / 6-

4 FOLLOWING THE FLOW Figure 3. The general diagram for tracking the flow of a conserved quantity. The model equations will have the general form: Top model} {Reservoirs} d/dt (Reservoir_1) = - Flow_12 INIT Reservoir_1 = 1 d/dt (Reservoir_2) = Flow_12 INIT Reservoir_2 = 1 {Flows} Flow_12 = Conductance*(Driving_Force_1-Driving_force_2) {Functions} Conductance = 1 Driving_Force_1 = Reservoir_1/Capacity_1 Capacity_1 = 1 Capacity_2 = 1 Driving_force_2 = Reservoir_2/Capacity_2 EXAMPLE. Consider the following situation where water flows across a semipermeable membrane in response to an osmotic pressure difference. - 4 / 6-

5 Membrane Water Solute We want to keep track of the flow of solutes and of water. The conserved quantities are N = number of solute molecules and M = mass of water. However, it is more convenient to scale these quantities: # moles of solute: n [mol] = N/N A, where N A is Avogadro s number = 6x10 23 molecules/mol. Volume of water: V [cm 3 ] = m [gm]/ρ [gm/cm 3 ]. Since the density of water ρ ~ 1, the volume is numerically equal to the mass. The driving force for diffusion of solute across the membrane is Fick's law: Diffusive flux of solute = P (c 1 c 2 ), where c i = concentration of solute in compartment I = 1, 2, and P is the permeability. Molar concentration of solute: c [M] = n/v (M = moles/liter) The driving force for water flow is the pressure difference: Water flow = P V (p 1 p 2 ), where the pressure difference is computed from van t Hoff's law: (p 1 p 2 ) = RT (χ 1 - c 2 ). The Flowchart for this problem is shown below. As the Flowchart is being constructed, the equation window shows the model equations being assembled. - 5 / 6-

6 {Top model} {Reservoirs} {Flows} {Functions} d/dt (N1) = - J12 INIT N1 = 1 d/dt (N2) = J12 INIT N2 = 0 d/dt (V1) = - Jv INIT V1 = 1 d/dt (V2) = Jv INIT V2 = 1 J12 = Permeability*(C1-C2) Jv = Pw*(P1-P2) Permeability = 1 C1 = N1/V1 C2 = N2/V2 Pw = 1 P1 = RT*C1 P2 = RT*C2 RT = 1-6 / 6-