Engineering Problem Solving as Model Building

Engineering Problem Solving

as Model Building

Part 1. How professors think about problem solving.

Part 2. Mech2 and “Brain-Full Crisis”

Part 1 How “experts” think about

problem solving

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 sub-models, assumptions and conservation principles.

Possible model structure for

many thermodynamics

problems

Control mass or control volume drawing, boundaries usually chosen where things are known or desired

Process: Rev., Irrev.,

adiabatic, PVn_=constant..

State diagram (P-V, T-S…) (optional but usually VERY helpful)

Property Model

Ideal gas, incompressible liquid, real gas, 2-phase

Thermodynamic Relations

H=U+PV, dU=TdS-PdV, Cp= ….(puts variables in more convenient forms) State Change, E₂-E₁= …, S₂ -S₁=… Other Physics Mechanics, heat transfer theory, ….

Problem Statement (specify known, unknown desired quantities, and possibly some assumptions to fill gaps)

Mass, Energy Conservation Entropy Balance

This is not a set of directions!

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

A mass M of steel is heated from T₁ to T₂, there is heat transfer Q to the steel, and work W by the steel.

Variants of the problem:

1. M, T₁, T₂ given, find Q, W 2. Q, M, T₁ given, find T₂ 3. Q, T₁, T₂ given, find M 4. M, T₁, T₂ given find S₂-S₁ Steel Mass M Q

Control mass

there is no flow, and it is sensible to take the same system (the steel) for all problem variants

Steel Mass M

Q

W

Mass conservation is trivial (M=constant) Energy conservation is E₂-E₁=Q-W

Entropy Balance is dS=δQ/T +dS_gen

Assumptions

• 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.

Steel

Property Model

• Simple compressible substance (only boundary work is possible, and it is zero in this case)

• v=constant even if T, P change so
(Cp=Cv=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

Steel

Mass M Q

Process Information

• Constant volume, so W=0

• No information to suggest Q is zero, so it

must be retained in 1st _Law

• May or may not be reversible, so unclear if we can relate Q to entropy

Steel

Other Physics

• 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.

Steel Mass M

Q

Thermodynamic Relations

• 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=Tds-Pdv or ds= du/T=CdT/T

Steel

The complete model

U₂-U₁=Q-W but W=0 and U related to T

MC(T₂-T₁)=Q

• for problems #1, 2, 3, use trivial algebra. • for problem #4, we also need to integrate

dS=CdT/T Steel Mass M Q

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. The diagrams reinforce important parts of the model related to our property model and the path.

Steel Mass M

Control mass drawing

Process:

Constant V so W=0

State diagram (P-V, T-S…) (optional but usually VERY helpful) Property Model V const; u(T), s(T) Thermodynamic Relations dU=TdS-PdV, du=CdT (const. V) State Change, E₂-E₁=U₂-U₁=MC(T₂ -T₁) S₂-S₁=MC ln (T₂/T₁) Other Physics

Seems KE, PE not relevant

Problem A mass M of steel is heated

from T1 to T2, there is heat transfer Q to the steel, and work W by the steel. M, T1, T2 given, find Q, W

Mass, Energy Conservation E₂-E₁=Q-W

Entropy Balance dS ≥δQ/T Steel Mass M W Q

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.

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

Thermo Lectures 1-3 PVT

Properties

3 Model Building Blocks _{3 Complete Models}

Ideal gas Ideal gas

Steam Tables _{Steam Tables} Incompressible

liquids and solids

Incompressible liquids and solids

Thermo Lectures 1-9 PVT, Energy

and First Law

3 Major Model Components,

Perhaps 9 Sub-models _{3x2x4=24 Complete Models}

For example, just using First Law in Integrated form, 12 models:

Ideal gas

Steam Tables liquids and solids

E₂-E₁=Q-W or rate form Const V Cylinder Insulated vs isothermal Const V Const V Const V Const V Const V Const V Cylinder Cylinder Cylinder Cylinder Cylinder Cylinder

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

Add all the rest

• Control volume analysis • 2nd _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?

Over the weeks…

time

# complete models

# model building blocks

Your brain capacity Things to

remember

Over the weeks…

time

# complete models

# model building blocks

Your brain capacity Things to

remember

Best test scores by memorizing examples

Over the weeks…

time

# complete models

# model building blocks

Your brain capacity Things to

remember

High-school

First Year UBC Mech 2

Brain Full Crisis

Have you reached Brain-Full Crisis

(BFC)?

We’ve given you mixed messages

not exactly like past examples • Given time-limited 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 long-term investment.

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 unlimited-time 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

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 T₁=25 C, and the

pressure is P₁=120 kPa.

At the outlet (3), P₃=50,000 kPa and T₃=25.6 C

All parts of the pump, piping and flow resistance are well insulated. The fluid is diesel with

density ρ=820 kg/m3_{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.

1

2 3

Ideal pump Flow resistance

Control volume Process: State diagram Property Model Thermodynamic Relations State Change Other Physics Problem

Mass, Energy Conservation

Entropy Balance

1

2 3

Ideal pump Flow resistance

Shaft work

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.

RTT
CV analysis

1st_{Law control mass}

problems Ch. 4

1st_{Law CV}

problems Ch.5

1st₊₂nd_{Law problems Ch. 7}

Mech222 Notes Text/Notes _{Cengel &Boles}

Zeroth Law Equilibrium state,

PVT exist

Property Models (Ch. 3) Ideal gas, tables…

dS_univmax at equilibrium

equality of temperature V S U T ∂ ∂ ≡ T Q dS≥δ dE=δQ-δW Conservation of Energy Existence of E ∫ ≥ T Q δ 0 T Q dS≥δ T res. E W Q

Simple heat engine/pump Problems Ch. 6 η=1-T_L/T_H T res. E W Q η=1-TL/TH dE=δQ-δW

Given a few properties, calculate others (Ch. 3)

The “road map”…

• Explains how ideas depend on previous material.

• Compares approaches of text vs. notes • Is unrelated to how we normally solve

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, 1st _{Law, 2}nd _{Law …)?}

9. Solution method needed? Do you need to iterate….?

Problem Solving Method (CB 1-12)

Textbook problem solving steps

• comforting step-by-step 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.