Individual Report 2 Guidelines: • • • • This is an individual report (one submission per student). Use 12 pt Times New Roman, double-spaced. Recommended length: 3-6 pages. When completing the peer reviews, use the rubric below to assign a score to each of the five (5) subsection categories in the first column of the rubric. Sum the scores for each subsection to determine the total score for this assignment out of 5 points. For example, if you rate each subsection Proficient (0.75 pts per subsection), then the total score is 0.75 x 5 = 3.75 out of 5 possible points. Rubric: Subsection 1. Description of Physical System and Experimental Methods 2. Mathematical Model 3. Simulation Method Exemplary (1) Student’s descriptions of the testing system and experimental methods are concise yet thorough. The description of methods is detailed enough that one could replicate it, yet at the same time is not excessively wordy. Any photos or figures include captions. All equations describing the system dynamics are reported accurately and all variables are defined. Description of the simulation method reflects understanding of the physical meaning of the dH/dt ODE as well as the rationale behind using numerical solution methods (“ode solvers”). Proficient (0.75) Student’s descriptions of the testing system and experimental methods are adequate; minor steps may be missing, but overall the documentation of methods is adequate. Any photos or figures include captions. Apprentice (0.5) Student’s descriptions of the testing system and experimental methods lack sufficient detail. Documentation of how data were collected is incomplete or sloppy. Any photos or figures do not always include captions. Novice (0.25) Student’s descriptions of the testing system and experimental methods are missing or barely mentioned. Any photos or figures do not include captions. Most equations describing the system dynamics are reported correctly and all variables are defined. Description of the simulation method reflects understanding of the physical meaning of the dH/dt ODE. Some equations describing the system dynamics are reported and most of the variables are defined. Description of the simulation method indicates the simulation code has been written correctly, but does not reflect much understanding of the physical meaning of the dH/dt ODE. Incomplete reporting of the equations and variables used to describe system dynamics. Description of the simulation method indicates the simulation code has not been written correctly or provides too few details. 1 4. Calibration Method 5. Validation Method Exemplary (1) Description of the calibration method reflects understanding of how model calibration achieves a better fit between simulation and observed measurements. The methods used to analyze the experimental data and calculate the model error are described clearly and concisely. Description of the validation method reflects understanding of how the application of the calibrated model to another testing system allows an evaluation of model performance. Provides a clear and concise explanation of how the calibrated model was used to produce a simulation for a new testing system, and how this was compared to measurements from the second testing system (error analysis). Proficient (0.75) Description of the calibration method reflects understanding of how model calibration achieves a better fit between simulation and observed measurements. The methods used to analyze the experimental data and calculate the model error are described, but not always clearly and concisely. Description of validation method includes all information necessary to understand how the results were produced but is not always clear or concise. Apprentice (0.5) Description of the calibration method indicates that the model was calibrated correctly, but does not reflect much understanding of how the calibration algorithm works. The methods used to analyze the experimental data and calculate the model error are described, but some steps are missing. Description of validation method is missing some key points about how the calibrated model was applied to a new testing system and how model error was determined. Novice (0.25) Description of the calibration method indicates the optimization code has not been written correctly or provides too few details. The methods used to analyze the experimental data and calculate the model error are not described or are not covered in enough detail. Description of validation method provides too little explanation of the way that the calibrated model was applied to a new testing system. 2 Trial_No 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 H_cm 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13 12 11 10 9 Time_s 10.2 15.9 21.3 27 32.7 38.3 43.7 51.1 56.9 60.5 72.3 79.3 88.4 97.5 106.8 118.9 131.6 150 10.4 15.8 21.4 27.3 32.6 38.9 44.8 51.3 57.7 64.8 72.2 80.3 87.2 97.6 107.6 118.1 131.8 150 12.2 17.4 23.5 28.8 34.4 40.9 47.8 54.1 60.1 67.6 Delta_t_s 0 5.7 5.4 5.7 5.7 5.6 5.4 7.4 5.8 3.6 11.8 7 9.1 9.1 9.3 12.1 12.7 18.4 0 5.4 5.6 5.9 5.3 6.3 5.9 6.5 6.4 7.1 7.4 8.1 6.9 10.4 10 10.5 13.7 18.2 0 5.2 6.1 5.3 5.6 6.5 6.9 6.3 6 7.5 Time_since_start_s 0 5.7 11.1 16.8 22.5 28.1 33.5 40.9 46.7 50.3 62.1 69.1 78.2 87.3 96.6 108.7 121.4 139.8 0 5.4 11 16.9 22.2 28.5 34.4 40.9 47.3 54.4 61.8 69.9 76.8 87.2 97.2 107.7 121.4 139.6 0 5.2 11.3 16.6 22.2 28.7 35.6 41.9 47.9 55.4 3 3 3 3 3 3 3 3 8 7 6 5 4 3 2 1 74.3 82.6 90.9 100.4 109.8 121.3 136.8 153.3 6.7 8.3 8.3 9.5 9.4 11.5 15.5 16.5 62.1 70.4 78.7 88.2 97.6 109.1 124.6 141.1 Area_Water_Surface_cm2 90.47786842 90.16370916 89.84954989 89.53539063 89.22123136 88.9070721 88.59291283 88.27875357 87.9645943 87.65043504 87.33627577 87.0221165 86.70795724 86.39379797 86.07963871 85.76547944 85.45132018 85.13716091 90.47786842 90.16370916 89.84954989 89.53539063 89.22123136 88.9070721 88.59291283 88.27875357 87.9645943 87.65043504 87.33627577 87.0221165 86.70795724 86.39379797 86.07963871 85.76547944 85.45132018 85.13716091 90.47786842 90.16370916 89.84954989 89.53539063 89.22123136 88.9070721 88.59291283 88.27875357 87.9645943 87.65043504 87.33627577 87.0221165 86.70795724 86.39379797 86.07963871 85.76547944 85.45132018 85.13716091 Water System Modeling Project: Merging Theory and Data ENGRCEE 170 - Introduction to Fluid Mechanics October 25, 2021 1 Introduction 1.1 Project Rationale There is a major trend in civil and environmental engineering involving a transition from theory-driven analysis and design to data-driven analysis and design. For engineering students to be prepared for successful careers over the next 4-5 decades, an ability to synthesize theory and data using models (including custom-made models) will be paramount. This project gives students an introduction to this process. It covers a set of essential topics: 1. Collection of data using physical systems useful for developing and testing theories and analysis of measurement error. 2. Formulation of mathematical models based on engineering mechanics fundamentals. 3. Development of software that uses numerical methods to solve the mathematical equations and arrive at quantitative solutions. 4. Development of software that synthesizes theory and measurements and yields information about the uncertainty in quantitative solutions. 5. Making a prediction from a model that is grounded in both theory and data. 1.2 Pre-Requisites Students should have the ability to use programming tools (e.g., Matlab, Python, C, Fortran) for this project, and should also be familiar with solving ordinary differential equations. The pre-requisites of this course are designed to check for these qualifications. If you haven’t previously taken a programming course but are enrolled in one now, I am willing to sign a pre-requisite waiver. 1.3 Project Overview The project will be completed over the first eight weeks of the quarter, with the aim of leaving the final two weeks to focus on preparation for the final exam. Class projects invariably push students to work very hard and invest large amounts of time, which is good, but not when it takes away from time that students can and should be using to solve lots of problems and understand all important concepts in a course. This project will involve the collection of data using a very simple water tank system, the synthesis of theory and data through modeling (and programming), and the testing of theory through making a prediction. Importantly, the project will provide students a hands-on introduction to modeling and simulation, which is relied upon extensively today by engineers for analysis and design purposes. 1 The main stages of the project are as follows: 1. Experiments yielding water level and discharge data. 2. Writing a code that simulates system dynamics. 3. Writing a code that calibrates parameters to synthesize theory and data. 4. Making a prediction, testing it against measurements, and analyzing errors and uncertainties. 5. Report preparation. Each step of the project is described in detail in Section 3. 2 Theory 2.1 Mathematical Modeling The integral statement of mass conservation is given by Z Z ∂ ρ d− V + ρ u · n dA = 0 ∂t CV CS (1) and if we define S to be the volume of water stored in the tank, Q to be the volumetric flow rate of water leaving the tank, and ρ to be a constant, then Eq. 1 can be re-written as follows, dS +Q=0 dt (2) Furthermore, if the tank has a cross sectional area, AT (z), where z is the vertical coordinate relative to a fixed datum, and we assume that the water height above a fixed datum is given by H, then Eq. 2 can be written as follows, AT (H) dH +Q=0 dt (3) Application of Torricelli’s law leads to an analytical expression for the velocity of water leaving a hole in the side of the tank located at a height zo above the fixed datum. This function appears as a function of the height of the water H as follows, p V (H) = 2g(H − zo ) (4) and we note that Torricelli’s Law is simply an application of Bernoulli’s equation between the surface of the tank and the outlet hole of the tank. The volumetric flow rate Q is related to the V through the area of the hole, Ao , and a discharge coefficient cD , as follows, Q = cD Ao V (5) The discharge coefficient is unity for an ideal fluid, but working with real fluids introduces viscous effects that lead to non-uniformity in the distribution of velocity out of the hole as well as friction losses. Hence, the discharge coefficient accounts for differences between ideal flow theory and real fluid behavior. Moreover, we generally expect that the discharge coefficient will be close to unity, and most 2 likely between a value of zero and unity. Now, by combining Eqs. 3, 4, and 5, and rearranging, we arrive at a mathematical model of the water level change in the tank over time,   p dH Ao = −cD 2g(H − zo ) (6) dt AT (H) with the initial condition H = H0 at t = t0 . Note that the mathematical model given by Eq. 6 and its initial condition represents an ordinary differential equation (ode), where t represents the independent variable and H is the dependent variable. Further, the parameters of the equation are cD , g, zo , Ao and there is one parametric function, AT (H). This ode can be solved analytically in some cases, such as when AT is a constant, cD is constant, and zo = 0. On the other hand, numerical solutions are almost always possible irrespective of the values and/or functional dependence of parameters. 2.2 Numerical Solution Methods There are a class of methods for integrating ode’s that are described as ode solvers. These methods assume that any system of ordinary differential equations (i.e., a set of n equations describing the time-wise change of n different dependent variables) can be written as follows, dy = f(t, y; p) dt (7) where y is a vector of dependent variables of dimension n, f a vector-valued function of dimension n describing the time rate of change in y, and p is a vector of parameters of dimension m needed to express f in based on values of t and y. Moreover, this system requires an initial condition given by y = y0 at t = t0 . When we apply this general formulation of ode’s to our specific problem, we see that Eq. 6 involves one dependent variable (n=1) and at minimum five parameters (m=5): g, Ao , AT , zo and cD . Computers integrate ode’s by repeating a very simple cycle over and over: estimating y a short distance into the future, time t + ∆t, based on the value of solution at time t and the value of dy/dt at time t. Mathematically, this can be written as follows, y(t + ∆t) = y(t) + dy(t) ∆t dt (8) which corresponds to an algorithm named Euler’s Method. Now it is helpful to switch notation into something that is more compact and easier to write. Instead of writing out y(t), we’ll just use a subscript i to refer to time t, and i + 1 to time t + ∆t. Hence, Eq. 8 becomes  yi+1 = yi + dy dt  ∆t (9) yi+1 = yi + f(ti , yi ; p) ∆t (10) i Moreover, if we combine Eqs. and we can write and we can now create a simple Matlab algorithm that repeats this calculation over and over again to solve a very simple ode: dy/dt = −e−at from t=0 to 10 s for a=0.5 s−1 subject to y = 10 at t = 0 3 clear; close all a=0.5; T=10; dt=0.1; N=floor(T/dt); y(1)=10; t(1)=0; for i=1:N f=-exp(-a*t(i)); y(i+1)=y(i)+f*dt; t(i+1)=t(i)+dt; end plot(t,y) The advantage of using Euler’s method is that it is very simple, which makes it easy to code. However, it isn’t very accurate. Its accuracy is characterized as “first order”, which means that every time ∆t is cut in half, the truncation error is cut by a factor of two. Hence, there is a one-to-one proportionality between ∆t and errors. To get much smaller truncation errors, we will use a routine built into Matlab that achieves “fourth order” accuracy, which means that every time ∆t is cut in half, the error is cut by a factor of two to the fourth power!, i.e., 24 = 16. To use the built-in Matlab function, it is useful to first create a function that evaluates the right-handside of the ode. For the simple problem considered here, this can be written as: function f = odefcn(t,y,a) %Note that y is not used, but it is a requirement and it %will be needed for the class project %Input variables: t=ind var, y=dep var; a=parameter %Output variable: f=dydt f=-exp(-a*t); end Once this function file is created, we create a script file (Matlab program) that calls the built-in ode solver as follows: clear; close all T=10; a=0.5; y0=10; tspan=[0 T]; [t,y] = ode45(@(t,y) odefcn(t,y,a), tspan, y0); plot(t,y) For this project, each student will need to create a function file to evaluate the right-hand side of Eq. 6 and then a script file (Matlab program) that computes H versus t. Moreover, once H is computed, students can compute V using Eq. 4 and Q using Eq. 5. Hence, students can create a program that computes H, V and Q versus t and supports comparisons between model predictions and measurements. 3 Project Steps 3.1 Experiments Yielding Water Level and Discharge Data 3.1.1 Experiment 1 In this first experiment, you will be collecting water level data to develop and calibrate a model of the discharge from your testing system over time. The testing system consists of a simple reservoir that drains through a single outlet. Before you begin this experiment, think about how the height 4 of water in the testing system might change over time based on your current understanding of fluid mechanics. Will the trend be linear? Non-linear? How does the change in flow rate over time relate to the change in water height over time? You will fill the testing system with water and allow it to empty via gravity for at least five trials during this experiment. Video recordings will allow you to conduct the experiment and transcribe water level observations over time from the video playback later. The experimental procedure for Experiment 1 will be described in detail during your in-person lab session. Don’t forget to take photos of the experimental materials and methods to include in your reports! 3.1.2 Data collection for Experiment 1 Once you have obtained the video recordings for at least three (3) experimental trials, you will need to record time and water level measurements from the videos. We recommend using the “Data Collection Template.xlsx” spreadsheet uploaded to the Canvas website to organize the measurements needed to calibrate your model of water level, velocity, and discharge over time. 1. Upload each video to cloud storage (such as Google Drive) or to a computer so that you can delete it from your recording device’s memory. 2. You will be reading time observations from the elapsed time of each video, so the measurements will be more precise if you play the video in a media player that displays time in fractions of seconds. One example is VLC Media Player, which is available for free for Windows, Mac, and Linux. VLC Media Player does not display the video time in milliseconds by default, but you can install a free extension called Time that allows you to display the time in milliseconds. If the default media player on your computer displays time in whole seconds you can still record the time measurements, they will just not be as precise. 3. Open one of the experimental trial videos in a media player on your computer. Open the “Data Collection Template.xlsx” spreadsheet and select the “TS1 H vs t” tab. 4. Fast forward the video until the water level in the testing system is at the greatest height reading on your water level scale (e.g. 18 cm). You are aiming for the point when the bottom of the meniscus (the curve of surface of the water) is at the desired marking on the height scale. The nice thing about having a video recording of the experiment is that you can move forward or backward in the video until you feel the water level is exactly where you want it to be on the scale, so feel free to adjust until you feel the water level reading is right. 5. Record the trial number (1, 2, 3, etc.) in the “Trial No” column and the water level in the “H cm” column of the “Data Collection Template.xlsx” spreadsheet. Now record the corresponding time elapsed since the start of the video in column “Time s”. This time represents the t=0 or the start time of the experiment. 6. Continue recording the video time and water level that corresponds to each centimeter on the height scale, up to and including the lowest line on the scale. 7. Once you are done entering the water level and time data into the spreadsheet, calculate the column “delta t” as the difference between consecutive time measurements (e.g. cell C3-C2, C4-C3, etc.). Note that the first value in the “delta t” column should be set to zero, since there are no time measurements preceding the first one. 8. Calculate the column “Time Since Start s” as the delta t value for a given row plus the Time Since Start s value for the previous row (e.g. D3+E2, D4+E3, etc.). The first value in the “Time Since Start s” column should be set to zero, indicating the start time of the experiment. 5 9. Once you are done entering the water level data into the “TS1 H vs t” tab, enter the geometry data for Testing System 1 that you recorded during lab into the “TS1 Geometry” tab of the spreadsheet, if you have not already. 3.1.3 Experiment 2 The experimental procedure for Experiment 2 is essentially the same as that for Experiment 1 (see Section 3.1.1), except you will be using a second testing system with a different shape than the first. If your second testing system has an irregular shape such that the cross-sectional area varies non-linearly with height, during this experiment you will need to pour multiple known volumes of water into the container and record the height that the water level rises to so that you can develop an equation that expresses the cross-sectional area as a function of height. This procedure will be described in more detail during your in-person lab session. The data collection procedure is the same as that of Experiment 1 (see Section 3.1.2), but make sure to populate the spreadsheet tabs labeled “TS2”. 3.2 Simulation Code To Be Determined. 3.3 Fitting Code To Be Determined. 3.4 Prediction, Testing, and Analysis of Errors To Be Determined. 6 4 Project Report Project grades will be based on a combination of a team-based final report grade and peer-assessed interim reports prepared individually by students. The project report grade will be weighed most heavily - preparation of written reports is a critically important task in both scientific research and professional engineering. Over the duration of the quarter, students will be tasked with preparing three individual interim reports that can be re-used and re-purposed by students to prepare the final report. The final report will be structured as follows 1. Executive Summary 2. Rationale and Objective (Individual Report 1) 3. Theory and Methods for Measurements, Modeling, Parameter Estimation and Prediction (Individual Report 2) 4. Calibration and Uncertainty (Individual Report 3) 5. Prediction and Error Analysis 6. References 7. Appendices (Codes and Data) 4.1 Executive Summary This is similar to an extended abstract of a paper, and is very popular today for reports prepared for broad audiences. A reader should be able to read an executive summary, not read any other part of a report, and get the most important information that you have to share. This section should quickly summarize the rationale and methods used for this project, and then transition into what was learned. Here, what you want to do is report findings and/or conclusions that are supported by the work contained within your report. This is not a place where you report findings unrelated to your work, and it is not a place to share opinions. For this assignment, I think it would be nice to include two graphics in your executive summary: a plot showing a calibrated simulation, and a plot showing a prediction by the calibrated model (against measurements). Recommended Length: 2 pages. 4.2 Rationale and Objective This project is an exercise in modeling and simulation, which is used widely in many fields of science and engineering today. First, students should introduce “modeling and simulation” to the reader in the context of engineering analysis and design. This will require that students do literature research on this topic, collecting some good sources of information and reviewing it, and then introducing this topic to the reader (in your own words, with citations). Secondly, students should explain how this project is designed to support hands-on learning about modeling and simulation (i.e., by taking measurements from a system, formulating and calibrating a simulation code, applying the code to make a prediction, and then taking additional measurements to see how well the simulation code performs). Third, this section should conclude with an objective or a set of objectives that are nicely framed by the preceding narrative. Organizing these ideas into a narrative that is informative, interesting to read, easy to follow, and specific about the final objective is the challenge here. 7 Recommended Length: 2-3 pages. This will largely depend on how much background information about modeling and simulation you wish to share with the reader. 4.3 Theory and Methods for Measurements, Modeling, Parameter Estimation and Prediction Unlike the previous section which is meant to intrigue the reader and get them excited about learning more, this section should be factual and straight to the point. There is no need to further engage the reader’s interest here. Rather, you want to explain as concisely as possible how you will carry out this project - the theory you will use, the data you will collect, and the steps of modeling that you will carry out. I suggest the following structure: 1. Physical System: Water Tank with Outlet 2. Experiments and Measurements (water level, volume, times, etc.) 3. Mathematical Model (equations describing system dynamics; no need to derive, just report and define all variables and parameters) 4. Simulation Method (integration of ode in Matlab) 5. Calibration Method (manual calibration, or an automated method if you choose) 6. Validation Method (explain that the calibrated model will be tested on a different physical system, and that you will evaluate the prediction by taking measurements that can be used in error analysis; explain the sources of uncertainty in the model including uncertainty in measurements, uncertainty in the calibrated parameters, uncertainty in the mathematical model; finally summarize the idea that this step is what allows us to test whether a model developed for one system can be successfully transferred to a second system, and if not, why). Recommended Length: as short as possible, but this may need to be 3-6 pages long to cover everything. 4.4 Calibrated Model and Uncertainty This is the first section where you will show results. Here, you will show graphics and tables that reveal the level of agreement between the calibrated simulation and the measurements. Here, you will also want to show measurement uncertainties and the sensitivity of the calibrated simulation to various inputs, most importantly the parameters that you calibrated. Recommended Length: 2-3 pages including plots and tables. 4.5 Prediction and Error Analysis This the second results section. Here, you will show how well the calibrated model simulated system dynamics in a second physical system for which the model was not calibrated. It will be important to once again show uncertainties. We don’t expect models to exactly pass through data, but we do expect that a good simulation will fall within uncertainties that are predictable based on measurement errors, model sensitivities, and uncertainties in model inputs and parameters. Recommended Length: 1-2 pages including plots and tables 8 es _2022_10_27_001815.PNG feic -arn arn. inst -qui TACI St 3 Submit your Individual Report 2 here as a PDE (one submission per student). This report represents the Theory and Methods for Measurements, Modeling, Parameter Estimation and Prediction section of your final report, which is described in Section 4.3 in the Project Description and Guide linked below: CEE 170 Project Description and Guide F2021.pdf+ Click the link below to download additional assignment guidelines and a rubric for Individual Report 2: Individual Report 2 Guidelines and Rubric F2021.pdf Peer Assessment Assignment: Your report will be peer-assessed by other students in the class (your comments are anonymous), and you will automatically be assigned the reports of three (3) other students to peer assess on October 28th at 11:59 pm PST. Use the Rubric above to rate your peers' reports out of 5 possible points and provide 1-2 sentences of constructive feedback explaining your rating **The peer assessment is due Friday. November 28th at 11:59 pm PST for full credit. How to access the reports you are assigned to peer review: On the right side of the screen, under your submission information, you should see "Assigned Peer Reviews." Click on one of the links that say "Anonymous User." • Write your peer assessment in the "Add a Comment" box. Make sure to assign a total rating out of 5 based on the above rubric and provide 1-2 sentences explaining the reason for your rating . Repeat for the other two students. • For more info on how to submit peer assessments, click here > Grading: .5 points for submitting Individual Report on time ⚫ 5 Points for completing peer assessments on time • 10% deducted for each day assignment is late from the appropriate score Note that this assignment will be reviewed for plagiarism using Tumitin. 8 x