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lec3_print - Examples of Dynamic Models Srinivas Palanki...

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Examples of Dynamic Models Srinivas Palanki University of South Alabama Srinivas Palanki (USA) Examples of Dynamic Models 1 / 19
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Examples Liquid Storage Process A typical liquid storage process is shown in the figure below. q q in Filling Process How does the height change with time ? Srinivas Palanki (USA) Examples of Dynamic Models 2 / 19
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Examples Step 1: Problem Statement Given in the previous slide Step 2: Assembly of information Sketch : q q in Filling Process System and Notation q in volumetric flow rate in, m 3 / s q volumetric flow rate out, m 3 / s V volume of liquid in tank, m 3 h height of liquid at time t , m A area of cross section, m 2 ρ liquid density, kg / m 3 C v flow coefficient, m 2 . 5 Srinivas Palanki (USA) Examples of Dynamic Models 3 / 19
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Examples Axiomatic Law : Conservation of mass Rate of accumulation of mass = mass flow rate in - mass flow rate out d dt ( ρ V ) = q in ρ - q ρ (1) Rate law and Constitutive Relations : V = Ah q = C v h (2) Assumptions : Density is constant (temperature is constant) No reaction in tank Srinivas Palanki (USA) Examples of Dynamic Models 4 / 19
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Examples Step 3: Combine d dt ( ρ Ah ) = q in ρ - C v h ρ (3) Step 4: Boundary Condition At t = 0 , h = h 0 (4) Srinivas Palanki (USA) Examples of Dynamic Models 5 / 19
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Examples Step 5: Simplify ρ A dh dt = ρ q in - ρ C v h h (0) = h 0 (5) This can be further simplified to: dh dt = 1 A q in - C v A h h (0) = h 0 (6) If h 0 , A and C v are known, the above equation can be solved to give a relation between the volumetric flow rate into the tank and the height of liquid in the tank.
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