Welcome W 10.1 Introduction to Engineering Design II (IE 202) Chapter 6: Modeling and Simulation 1 1

Today’s Learning Outcomes By the completion of today's meeting, students should be able to: Become aware of the idea of modeling and simulation. Identify the role of mathematical modeling. Perform homogeneity test for the units in equations. 2 2

Preview What have we covered so far? Understanding the client’s needs. Defining the problem in terms of objectives and constraints. Creating, expanding and reducing the design space. Coming up with a conceptual design. 3 3

3rd Task in the Design Process Where are we? 4 4

Studying the Chosen Design Goal: Verifying the success of the design chosen by examining how well it does the job. Methodology: Testing various design parameters, till optimum values are obtained. How could this be achieved? Think as a team, then present your idea. 5 5

Modeling & Simulation Usually modeling is used in the design process. Then simulation is done using … ? How is computer simulation helpful? Time; simulation tests are done in a second (depending on the complexity of the model) allowing the model to be refined and optimized. Money; it is true that computation time is not free but could not be compared to the cost of building a prototype and testing it. 6 6

Questions before Modeling Why do we need to perform modeling? What do want to find with this model? What is the difference between data and assumptions? How can we develop a modeling equation? How can we verify the model predictions? Can we improve the model? 7 7

Basic Concepts Abstraction: choosing the right level of detail when constructing the modeling equation. Scaling: weighing the elements used in the modeling equation. Abstraction depends on scaling each element in the original equation. Lumped Elements/Parameters: used to reduce the number of terms, and thereby simplify the modeling equation. Linearization: V = p h r2 linear with h, not r 8 8

Dimensions and Units Basic Units: time (T), length (L) & mass (M) As teams, find the units of the following Derived Units: force (F) = acceleration (a) = density (r) = velocity (v) = viscosity (m) = weight (w) = pressure (p) = sin q = stress (t) = modulus of elasticity (E) = 2nd moment of inertia (I) = 9 9

Dimensions and Units Ans. velocity (v) = (L/T) = (LT-1), sin q = (1) acceleration (a) = (L/T2) = (LT-2) force (F) = (ML/T2) = (MLT-2) weight (w) = (ML/T2) = (MLT-2) density (r) = (M/L3) = (ML-3) viscosity (m) = (M/LT) = (ML-1T-1) pressure (p) = (force/area) = (M/LT2) stress (t) = (force/area) = (M/LT2) = (ML-1T-2) modulus of elasticity (E) = (force/area)= (M/LT2) 2nd moment of inertia (I) = bh3/12 = (L4) 10 10

Dimensional Analysis All equations should be dimensionally consistent/homogenous. Check the dimensions of each of the following equations v2 = 2 g h , V = p h r2 , d = F L3 / C E I (P2 - P1) / r + g (h2 – h1) + (v22 – v12) / 2 = 0 What do you recognize? Can we get terms without dimensions? How can the dimensions cancel each other? Why would one develop dimensionless numbers? Much less calculations and experimentation to do homogenous By division To reduce the no. of variables. So what? 11 11

Practice Check the dimensions of each of the following equations dP / r + g (h2 – h1) + (v22 – v12) / 2 = 0 [M/LT2] / [M/L3] + [L/T2] [L] + [L2/T2] / 1 = 0 => L2/T2 + L2/T2 + L2/T2 = L2/T2 v2 = 2 g h => ([L/T])2 = [L/T2] [L] => L2/T2 = L2/T2 V = p h r2 => [L3] = [L] [L2] => L3 = L3 d = F L3 / C E I L = [ML/T2] L3 / [M/LT2] [L4] => L = L 12 12

Dimensionless Numbers Dimensionless Numbers: terms developed by scientists to express ratios of quantities indicating scientific understanding of the numbers. Dimensionless numbers are the terms used in dimensionless relationships. Find the dimensions of the following numbers Reynolds Number: Re = r g h / m Lift Coefficient: CL = FL / (0.5 r v2 A) Mach Number: Ma = v / vs Froude Number: Fr = v2 / g h 13 13

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