Engineering Problem Solving as Model Building


 Mavis Douglas
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1 Engineering Problem Solving as Model Building Part 1. How professors think about problem solving. Part 2. Mech2 and BrainFull Crisis Part 1 How experts think about problem solving
2 When we solve a problem using theory, we are (whether we realize it or not) constructing a model of the problem. Why model? A physical model boat is different from the real boat, but by pushing or pulling on the model, we can get information about the real boat (your foam boats taught you about stability and drag). A theoretical model is similar in that once it is constructed, we can use it to answer many questions. Construction of the model means selecting a consistent set of submodels, assumptions and conservation principles.
3 Possible model structure for many thermodynamics problems Problem Statement (specify known, unknown desired quantities, and possibly some assumptions to fill gaps) Control mass or control volume drawing, boundaries usually chosen where things are known or desired Mass, Energy Conservation Entropy Balance State Change, E 2 E 1 =, S 2  S 1 = Process: Rev., Irrev., adiabatic, PV n =constant.. Other Physics Mechanics, heat transfer theory,. State diagram (PV, TS ) (optional but usually VERY helpful) Property Model Ideal gas, incompressible liquid, real gas, 2phase Thermodynamic Relations H=U+PV, du=tdspdv, Cp=.(puts variables in more convenient forms)
4 This is not a set of directions! Arrows show that boxes are connected and consistent, not steps in problem solving. Together, modules (the boxes) make a complete model. From the model we get mathematical relations between the variables. The solution order depends on what we seek. A simple example Steel Mass M A mass M of steel is heated from T 1 to T 2, there is heat transfer to the steel, and work W by the steel. Variants of the problem: 1. M, T 1, T 2 given, find, W 2., M, T 1 given, find T 2 3., T 1, T 2 given, find M 4. M, T 1, T 2 given find S 2 S 1
5 Control mass there is no flow, and it is sensible to take the same system (the steel) for all problem variants W Steel Mass M Mass conservation is trivial (M=constant) Energy conservation is E 2 E 1 =W Entropy Balance is ds=δ/t +ds gen Assumptions Steel Mass M No information on elevation change or velocity, so neglect them. No information on the steel, so based on past problems, we might assume that it behaves like an incompressible and constant volume solid, with property information in textbook. Keep open to the possibility that later these assumptions are inconsistent with the other parts of the problem model, and therefore inappropriate.
6 Property Model Steel Mass M Simple compressible substance (only boundary work is possible, and it is zero in this case) v=constant even if T, P change so (C p =C v =C) u=u(t) s=s(t) Because these properties are independent of pressure, we may not need to worry about lack of information on P Process Information Steel Mass M Constant volume, so W=0 No information to suggest is zero, so it must be retained in 1 st Law May or may not be reversible, so unclear if we can relate to entropy
7 Other Physics Steel Mass M In some problems, we might need to relate applied forces to pressures in the system, solving equations of statics or dynamics. In some problems, heat transfer might be related to temperatures thought heat transfer theory. In this particular example, we need not worry about any such constraints because our system is a static, incompressible lump. Thermodynamic Relations Steel Mass M Text provides C (kj/kg/k), and the problem statement may involve temperature. The first law involves energy, so we need to relate, u, C, T: C=du/dT (for our case with the solid) du=tdspdv or ds= du/t=cdt/t
8 The complete model Steel Mass M U 2 U 1 =W but W=0 and U related to T MC(T 2 T 1 )= for problems #1, 2, 3, use trivial algebra. for problem #4, we also need to integrate ds=cdt/t State diagrams It has NOT been necessary to assume reversibility in this problem, so we DON T know for sure the path from 1 2. Steel Mass M The diagrams reinforce important parts of the model related to our property model and the path. T 2 T T T 1 v S
9 Problem A mass M of steel is heated from T1 to T2, there is heat transfer to the steel, and work W by the steel. M, T1, T2 given, find, W Control mass drawing Steel W Mass M Mass, Energy Conservation E 2 E 1 =W Entropy Balance ds δ/t State Change, E 2 E 1 =U 2 U 1 =MC(T 2  T 1 ) S 2 S 1 =MC ln (T 2 /T 1 ) Process: Constant V so W=0 Other Physics Seems KE, PE not relevant State diagram (PV, TS ) (optional but usually VERY helpful) Property Model V const; u(t), s(t) Thermodynamic Relations du=tdspdv, du=cdt (const. V) Experts vs Novices Experts tend to have a good framework or structure for their models, and are practiced in the art of assembling the model building blocks. Novices tend to focus on the final model, because it provides a fast way to compute answers.
10 Part 2. Why Mech 2 Brings you to the Point of Crisis Should you construct or memorize models? Construction Requires skills in math and very firm foundations Only memorize the building blocks Essential for new problems Not the fastest way to solve old problems Memorization Does not depend on foundations. Many, many models to memorize. Useless for new problems. Fastest way to solve old problems
11 Thermo Lectures 13 PVT Properties 3 Model Building Blocks 3 Complete Models Ideal gas Ideal gas Incompressible liquids and solids Incompressible liquids and solids Steam Tables Steam Tables Thermo Lectures 19 PVT, Energy and First Law 3 Major Model Components, Perhaps 9 Submodels 3x2x4=24 Complete Models Ideal gas liquids and solids Steam Tables For example, just using First Law in Integrated form, 12 models: Const V Const V Const V E 2 E 1 =W Const V Const V Const V or rate form Cylinder Cylinder Cylinder Const V Cylinder Cylinder Cylinder Cylinder Insulated vs isothermal
12 Add springs 1 more variation in model building blocks Each piston problem could now be with or without springs (insulated or not) Now 3x2x6=36 complete models Add possibility of piston kinetic energy 1 more variation in model building blocks Piston problems now insulated (or not), with spring (or not), with KE (or not) = 8 piston variants Total complete models =3x2x(2+8)=60
13 Add all the rest Control volume analysis 2 nd Law Machinery with many parts.. Steady vs transient problems Textbook has over 1000 problems! Fluids + Thermo + Math? In the first few years of mech 2, we set exam problems combining all 3 subjects. How many complete models to memorize? How do think students liked this?
14 Things to remember Over the weeks # complete models Your brain capacity # model building blocks time Things to remember Over the weeks Best test scores by memorizing examples # complete models Need to construct models Your brain capacity # model building blocks time
15 Things to remember Over the weeks Brain Full Crisis # complete models Your brain capacity Mech 2 First Year UBC # model building blocks Highschool time Have you reached BrainFull Crisis (BFC)?
16 We ve given you mixed messages Stressed importance of derivations, understanding Assigned model building MATLAB and physical labs Given quiz problems not exactly like past examples Given timelimited computational tests Assigned relatively few marks to complex, longer model building assignments The time to start practicing model construction is today. In studying for the finals Review and list the basic building blocks. Focus on how building blocks have been glued together in past problems. DO NOT spend time on new examples, except to test your model building. Remember that this is a longterm investment.
17 Discussion What sort of exercises would promote ability to construct models rather than just use them? What sort of testing would discourage memorization of problem solutions (this could influence how the final exams are set). Do you already have experience with constructing models from scratch, but in another part of your life? From the discussion after the lecture Should consider unlimitedtime exams to remove the incentive to memorize whole problems (this will take some work, but should be possible for some, if not all, exams). Exam marking schemes should clearly indicate (where appropriate) that most of the marks come from problem setup (ok we will check final exams for this) Vista problem sets might be set up to emphasize construction of models from building blocks (not sure how to do this, but it is worth considering)
18 Extra slides not covered in class (but probably worth a quick read) Another example: A diesel pump with friction might be thought of as an ideal, frictionless pump in series with a flow resistance (a throttling process). At the inlet to the pump (1), the mass flow is 0.2 kg/s, the temperature T 1 =25 C, and the pressure is P 1 =120 kpa. At the outlet (3), P 3 =50,000 kpa and T 3 =25.6 C All parts of the pump, piping and flow resistance are well insulated. The fluid is diesel with density ρ=820 kg/m 3 and heat capacity 2.0 kj/kg/k. Find the shaft work from the pump. Indicate your choice of control volumes carefully and explain any further assumptions needed. TRY THIS: TAKE THIS PROBLEM AND COMPLETE THE MODEL TEMPLATE ON THE NEXT PAGE. Ideal pump Flow resistance Shaft work
19 Problem Control volume Ideal pump 1 Shaft work Flow resistance 2 3 Mass, Energy Conservation Other Physics Entropy Balance State Change Process: State diagram Property Model Thermodynamic Relations Alternative connections between ideas Course concept road map showing the order topics covered (based how theory is developed) Components of the problem solving process given in text (and earlier notes) Thinking of problem solving as construction of a model rather than applying a problem template.
20 Mech222 Notes Text/Notes Cengel &Boles Equilibrium state, PVT exist Conservation of Energy de=δδw Property Models (Ch. 3) Ideal gas, tables Given a few properties, calculate others (Ch. 3) 1 st Law control mass problems Ch. 4 RTT CV analysis Zeroth Law Existence of E de=δδw ds univ max at equilibrium U T S V equality of temperature 1 st Law CV problems Ch.5 T res. E W δ ds T Simple heat engine/pump Problems Ch. 6 η=1t L /T H T res. E W η=1t L /T H 1 st +2 nd Law problems Ch. 7 δ 0 T δ ds T The road map Explains how ideas depend on previous material. Compares approaches of text vs. notes Is unrelated to how we normally solve problems.
21 Problem Solving Method (CB 112) 1. Physical layout of the problem? Make a sketch!. 2. What control mass do you choose? Show on sketch! 3. Initial state? 4. Final state? 5. Process: is any property fixed or otherwise specified? 6. What thermodynamic properties are convenient? Use these for a state diagram 7. What model do you use for the material of interest? 8. What laws are needed (mass, 1 st Law, 2 nd Law )? 9. Solution method needed? Do you need to iterate.? Textbook problem solving steps comforting stepbystep process Identifies some of the key concept blocks : process, states, property models. We don t always solve problems in exactly the order stated, even if we do hit all of the concept blocks.
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