Project TOUCAN A Study of a Two-Can System Prof. R.G. Longoria Update Fall 2009
Laboratory Goals Gain familiarity with building models that reflect reality. Show how a model can be used to guide physical design of experiments. Use concept of a state space model to show how analytical and/or simulation models can be used to design dynamic system performance. Apply concepts from uncertainty analysis when making model-based predictions.
Specific Objectives Determine parameters for orifice flow equation Develop validated model for a single can with water exiting orifice Model and simulate a two-can system, and validate using experimental study Use validated model to design for a specific performance specification (e.g., achieve this height of fluid in can 1 or 2 to within some margin).
Two-Can Model The two-can model is formed by two basic elements: Fluid storage in can is modeled as a C element quasi-static assumption constant-area tank pressure-volume (P-) constitutive relation Orifice flow is modeled by as an R element Steady-flow Bernoulli model simplifying assumptions pressure-flow (P-Q) constitutive relation Relation to ME 344 modeling concepts
One Can (Tank) Model The can stores potential hydraulic energy (we ignore inertial effects), and we can show that this is a capacitive element by recalling that, g 1 Pgh A C where we assume the can has a constant cross-sectional area, and C is the hydraulic capacitance, C A g
When is a can or tank a C? Fluid inertia effects are not significant. The area is constant. The fluid velocity profile is always uniform. These are not necessarily true statements. Opening R h h P P
The Orifice as an R Element The flowrate is directly related to the pressure drop across the orifice; i.e., flow variable In system modeling, we identify that the flowrate is a flow variable and pressure is an effort variable. This fact, together with the knowledge that the orifice involves dissipative processes allows us to model the orifice as a resistive (R) element. 2 Q Cc Ao P See Appendix at end of these slides for details. effort variable
The Orifice Relation (Optional) The flowrate relation we have derived is used extensively for orifice flow predictions in a wide range of applications. For example, it is commonly applied when predicting hydraulic valve flows. If the pressure becomes negative, then take the absolute value, 2 Q Cc Ao sgn( P) P This allows you to use this equation to model flowrate-pressure relationships even when the pressure is changing sign.
One Can Dynamic System Model We can piece the elements together using the mass continuity equation. We will only treat incompressible flow here so, becomes, m m m stored in out Q Q in out For the can system, Q Q in out where the volume stored in the can is:
One-Can Dynamic System Model Now, consider that a source of flow charges a can, and it discharges freely. Using continuity, d dt Q Q Q in out We can select the volume as a state variable for this system. We need to show that the exiting flow can be expressed as a function of the volume state as well. The input flow is a known input.
One-Can Dynamic System Model The exit flow is estimated using the orifice flow relation, 2 Q Cc Ao P If we use the constitutive relation for the can element, g 1 P gh Then, A C d 2 1 Qin Cc Ao Qin K dt C is the state equation for the one can system.
h One-Can Experiments Case 1 If Q in is adjusted so it exactly equals Q out Q in Q out constant d dt Q Q Q in Q out in, e out, e This system is said to be in equilibrium or steady-state, and the volume is constant. 0 In general, set: dx dt f ( x, u, t) 0 e e Defines n algebraic equations that give you equilibrium conditions.
One-Can Experiments Case 2 Q in = 0. d dt 0 Q d out or K dt olume is a dynamic state. h Solution: Q out constant K ( t) o t,0 t T 2 2 e 1 0.8 o Ko 2 0.6 2 t t eq 1e 0.4 0.2 K 1 1 d K dt Nonlinear can o 0 0 0.5 1 1.5 2 t Linear cans never empty d Keq dt
Finding K from Experiments Now let s use Case 2 to design an experiment that allows us to determine the unknown parameter, K. The equation, d dt K is a nonlinear ordinary differential equation that can be integrated from the form, d Kdt
Model-Based Experiment (cont.) An experiment begins with an initial volume in the can and allows it to empty. This is expressed in the model by, 0 Te d Kdt o where the initial volume and time to empty are easily measured in the laboratory. 0 T o e initial volume time for initial volume to empty
Integration of the equation, yields, or, 0 o d Model Analysis 2 o KTe This relation suggests we should measure volume and time to estimate the parameter K. 0 o d 2 T e 0 Kdt Te 1 2 0 T Kdt K t 0 o 0 e
Data Collection and Analysis The relation, 2 o KTe suggests development of a simple linear model relating initial volume and time to empty. o K T 2 e Mirror-image the data to force data through the (0,0) point. o slope T e K 2 The model should force a (0,0) point - this is realistic!
What do we have? An experimentally calibrated can that we can fill to some volume and predict reliably what the volume will be and when it will empty. If I asked you to determine an initial volume of water in the can so that it emptied at some T e and did so within some margin of error, how would you do this?
We can build a two-can system The state equations for the two can system are found by simply applying continuity again for a second can. In doing so, you find the state equations for the volumes in cans 1 (top) and 2 (bottom) are, respectively, d1 K1 1 dt d2 K1 1 K2 2 dt where the K values are distinct for each can.
Cascaded vs. Coupled Note that these equations describe two cascaded cans in which the top can (1) freely discharges into the bottom can (2). Cascaded tanks This system should be distinguished from a two-tank system in which the tanks are connected by a pipe. The tanks in this latter case are coupled in the sense that the states are interrelated. Can 1 in the two-can system to be studied here does not depend on the state (volume) of can 2. Coupled tanks
Solution for Two-Can Model Consider the case where you fill the top can (1) and the bottom can (2) is empty, then you release the flow from the top can into the bottom can. You can t integrate the equations analytically a simulation is required. If I asked you to determine an initial volume of water in the top can so that the bottom can peaked at some T max and did so within some margin of error, how would you do this?
Laboratory Work You need to find the K values for two cans. You need to show the TA that you can predict the time to empty to within some margin of error, and you must demonstrate with experiment. You should convince yourself that your model for a two-can system can predict critical values such as peak heights, and that your time predictions match as well. You need to research a way to solve the second problem for the second week of lab. You ll be expected to come to lab prepared to run additional experiments, to show the TA what you came up with, and then to demonstrate the results with the physical experiment. In the next lecture, we ll discuss uncertainty analysis, which may be helpful for the second week s work.
Summary The one and two can systems rely on physical modeling of hydraulic system elements, and introduce the application of state space modeling. The one-can system provides an example of how an experiment can be designed from a system model. The experiment(s) can then be used to determine system parameters (e.g., K). The problems posed have been solved using nominal values. Sometimes predictions fail. Blame is usually assigned to shoddy lab work. Introducing the expression margin of error is intended to have us seek ways to assign blame in a more systematic manner.
Appendix Derivation of the orifice flow equation and coefficient
Orifice flow model 2 Use mass continuity and the steady Bernoulli equation: 0 t C d CS da P P 2 2 to show that: 2 2 1 1 2 2 gh1 gh2 1 2 2 2gh A A 2 1 What is A 2? Use the jet area. If there is an orifice where the jet area is not equal to the orifice area, we use a contraction coefficient.
Contraction Coefficient The contraction coefficient is defined by, C c Some common experimentally determined values: A A jet o From Munson, Young, and Okiishi (1990)
elocity Coefficient Using the contraction coefficient, our model for the exiting mass flowrate is, m Q A C A C jet c o v ideal where we ve used the velocity coefficient, C v actual ideal velocity coefficient which accounts for friction effects.
Ideal Orifice Flowrate Model Use the ideal velocity from the Bernoulli equation, 1 ideal 2 2 use gauge pressure, P, at the bottom of the can, and assume constant density, so 2gh A A 2 1 Q m C C A 2P 2P K A c v o o o 2 2 1 Cc Ao A 1
Ideal Orifice Flow Coefficient Take the contraction and velocity coefficients at 1, K o,ideal C C c v 2 2 2 1 Cc Ao A 1 1 Ao A 1 This allows an estimate of the flowrate given just simple measurements from a can; i.e., 2P 2 Q K A K A g K ideal o,ideal o o,ideal o ideal A1 where, K ideal K o,ideal 2gA A 1 2 o 1