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Engineering Equation Solver (Thai)
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Engineering Equation Solver (Thai)



This slides are used in the class "Skill and Personality Development for Mechanical Engineers".

This slides are used in the class "Skill and Personality Development for Mechanical Engineers".



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    Engineering Equation Solver (Thai) Engineering Equation Solver (Thai) Presentation Transcript

    • Engineering Equation Solver Denpong Soodphakdee, Ph.D. Department of Mechanical Engineering Khon Kaen University denpong@kku.ac.th
    • EES ME KKU License 2
    • What is EES? o EES (pronounced “Ease”) is a general purpose equation solver, modeling and analysis tool which has started life specifically for the purpose of engineering education o It is quite capable (it is also used in industry) and is more than adequate for engineering education purposes o Students find it far easier to use than any other software they have been introduced to, including:  Mathematica  Matlab  Mathcad 3
    • Advantage of EES o It requires no real programming (although you can!) o Implicit (iterative solver) – equations in any order o It is geared towards engineering problems o Units enabled and unit conversion routines o Formatted equations view with Greek letters and maths symbols o Lots of online example programs o Excellent online help and online manual o It comes FREE to the entire Department – BOTH students and staff! o Students can take it home – it is small in size! 4
    • Features of EES o Excellent engineering features:  Lookup tables with linear-, cubic- and quadratic interpolation  Regressions  Plots and overlay plots  Diagram window (User Interface)  Animation (Cool!)  Built-in property library - thermo, fluid and material properties (easily extendible by users)  Predefined engineering constants o Excellent engineering analysis features:  Parametric studies  Uncertainty propagation  Min/Max. 5
    • Solving Nonlinear Equations o How would you solve the following? x 2  y 3  77 x 2 y 1 2   x  1.234 o And an implicit equation in f such as the following?  D  1 2.51  2.0log     3.7 Re f    f 6
    • Tutorial 1 :: Solving Nonlinear Equations o Create a new EES worksheet and save it as BasicEquation.ees o Now type in the nonlinear set of equations and solve for the 3 unknowns x 2  y 3  77 x 2 y 1 2   x  1.234 o The order in which the equations are entered does not matter at all! use Ctrl+F to see the equations in formatted view 7
    • Equation Formatting o Two types of comments:  Comments in quotes are shown in formatted view  Comments in curly brackets are not shown in formatted view quot;Equation Formattingquot; – this will be shown in formatted view quot;!Equation Formattingquot; – this will be shown in red {Equation Formatting} – this will not be shown  Can also highlight any text (select and then right-click) 8
    • Equation Formatting o Ordinary variables and equations  quot;Define some variables. Actually, they are really constants as you cannot later assign other values to any of them!quot; a = 1 b = 2 c = 3 e = 4  quot;!A more complex equation using these variablesquot; sqrt(1 + (a+b)/c + d) = e • Look at the formatted view! Ctrl-F • Note the position of the unknown “d” in the equation - it does not have to be on the left! 9
    • Equation Formatting o Raising the power k^2 = 5 Exponents are shown as superscripts in format view. o Clever Greek letters! or DELTAT = 1 deltaP = 2 or OMEGA = 100 omega = 100 or THETA = 45 theta = 45 Note: Although the formatted view distinguishes between upper and lower cases, the EES solver does not! Hence “OMEGA” and “omega” are regarded as the same variable! 10
    • Equation Formatting o General formatting y_old = 10 quot;Subscriptquot; z|alpha = 9 quot;Superscriptquot; x_dot = 10 quot;It understands dots & double dots!quot; x_ddot = 2 quot;Double dotquot; x_hat = 2 quot;Hatquot; x_bar = 22 quot;Over barquot; angle|o = 20 quot;Superscriptquot; T|star = 325 quot;Special superscript - starquot; Y|plus = 0.12 quot;Special superscript - plusquot; T_infinity = 25 quot;Often used to denote freestreamquot; 11
    • Constant o EES defines a large number of constants. Check out Options > Constants. Of interest are the following: (gravity)  g# So one can write F = m * g# Instead of g = 9.81 [m/s^2] F=m*g (Stefan-Boltzmann constant – radiation)  sigma# (Speed of light)  C# (Universal gas constant)  R# So the Ideal Gas Constant for air would be: R_air = R# / MolarMass(Air) 12
    • The Unit System o EES is fully unit-aware o The Unit System is the first thing that should be set at the start of a project  Set from the Options menu (next slide)  Safer to explicitly set units using directives (which will override dialog settings): $UnitSystem SI MASS DEG KPA C KJ 13
    • The Unit System o The unit system can be set by Option > Unit System 14
    • The Unit System o Individual constants can be assigned units: m = 25 [kg] a = 2.5 [m/s^2] F=m*a o Units cannot be assigned for equations, but EES will automatically determine the units for F (shown in purple in the results window) 15
    • The Unit System 16
    • The Unit System o EES also allows unit conversions  Suppose we have the equation F = m a, but we want F in kN. If we set [kN] for F in the units map, we will get a warning So we do this: F = m * a * convert(N, kN) quot;Alternatively you can do this, but then you need to know the conversion constantquot; F_1 = (m * a) / 1000 [N/kN] F_2 = m * a * 0.001 [kN/N] 17
    • The Unit System o We can even convert between British Gravitational and SI units: m_3 = 10 [lbm] a_3 = 3.5 [m/s^2] F_3 = (m_3 * convert(lbm, kg)) * a_3 18
    • The Unit System o We can also assign units to constants in situ to make a constant clearer, for example: quot;This is clearer than the next...quot; time = 3.5 [h] * 3600 [s/h] quot;The fact that this is 3.5 hours is not as apparent!quot; time = 12600 [s] o EES online examples:  Examples/Units conversion/Checking units and unit conversion (HeatEx.EES) 19
    • Built-in Functions o EES provides built-in functions in the following categories:  Mathematics  Fluid properties  Solid / Liquid properties  EES Library routines  External routines o Example code can be pasted o Function Info (Help) 20
    • Built-in Functions o A Maths example x=cos(Value) quot;This is exactly as it was pastedquot; Now it is up to you to modify the statement as you want it. Maybe you wanted to do the following: theta = 30 [deg] x_coordinate = cos(theta) or z = cos(33) quot;Hardcoding values is rarely a good ideaquot; 21
    • Built-in Functions o Maths examples LogValue = log10(100)quot;The log10(Value) was pasted!quot; T = 140 [C] quot;Note American spelling!quot; E = E_(Aluminum, T) o Integral equations  EES can perform numerical integration and differentiation. How would you solve the following? 3 y   x dx3 0 quot;An integral equation – be sure to switch off complex numbersquot; y = Integral(x^3, x, 0, 3, 0.06) 22
    • Built-in Functions o Property examples For properties one typically has to specify conditions such as pressures and temperatures. Furthermore, one has to specify the material (a solid or a fluid). The simplest example is probably the density of a gas. Let’s paste the density for air from the Fluid Properties Function Info dialog: rho_1=Density(Air,T=T_1,P=P_1) 23
    • Built-in Functions o Solid property example quot;Young’s Modulus – note the underscorequot; T = 140 [C] E = E_(Aluminum, T) 24
    • Built-in Functions o The property functions can be pasted from menu Option > Function Information 25
    • The Option Menu o Have a careful look at the functionality provided under the Options menu: Variable Info  Function Info  Unit Conversion Info  Constants  Unit System  Stop Criteria  Default Info  Preferences.  26
    • Parametric Study o A parametric study is in essence the study of the influence of variations in one or more variables (parameters) on the solution. o In most software, a parametric study is performed by repeatedly solving the model whilst making adjustments to the desired variables (parameters) in the form of a loop. o EES accomplishes this very elegantly by using a spreadsheet-like approach. 28
    • Parametric Study Example o Let’s look at a really simple example  Say you want to perform a calculation such as: y  cos   But you want to perform this operation for several angles, say between 0 and 360 degrees.  To do this in EES, simply enter this equation in the equations window 29
    • The Parametric Table o A really simple example…:  EES does this in a particularly elegant way. It uses a spreadsheet to specify the variables that are to be specified as well at the variables for which the results are to be monitored: • The number of runs • Each row is a new run • theta is now specified in the table, and EES will automatically list the results of y in the same table 30
    • The Parametric Table o The independent (specified) variables are simply typed into the EES parametric table. One can manually type in all the values, or utilize the quick-fill button: 31
    • The Parametric Table o The simple equation can be solved for each value of theta and the results y are displayed in the table. 32
    • Plot Basics o Engineering data is often best visualized by means of graphs (plots). o Plotting in EES is really easy. Once the data is available, a plot can be generated in the following simple steps:  Select the plot type from the menu (e.g. X-Y)  Select the data source (e.g. Parametric table or array)  Select the dependent (Y-axis) and independent (X-axis) variables for plotting  Select the plot formatting: Heading and description • Line type and appearance (e.g spline, dot-dash, colour) • Marker and legend, tics, grid lines, number format • Automatic update from data source (on/off) • Scale of axes, log or linear plot type etc • 33
    • Plot Basics o Create x-y plot from Plots menu: 34
    • Plot Basics 35
    • Tutorial 2 :: Projectile Parametric Table o Lets create a more realistic model on which we can do a parametric study (Projectile ParametricTable.EES):  A simple projectile movement is used to demonstrate the use of a parametric study.  We can modify the angle theta as well as the initial velocity u either individually or simultaneously and determine their influence on the maximum distance that the projectile will travel. 36
    • Tutorial 2 :: Projectile Parametric Table o Equations of motion v=u+a*t s = u * t + (1/2) * a * t^2 o To calculate the maximum distance, calculate the time the projectile needs to reach maximum height by applying the first equation to the vertical velocity component (v = 0 and a = g). The total time will be twice this amount. o Now apply this total time to the horizontal velocity (which remains constant) using the second equation. The x-acceleration in the second equation is obviously zero.quot; 37
    • Tutorial 2 :: Projectile Parametric Table o So the equations will be as follows (remember the unit system!): $UnitSystem SI MASS C KPA KJ DEG quot;Equations of motion v=u+a*t Eq. 1 s = u * t + (1/2) * a * t^2 Eq. 2quot; quot;Define initial valuesquot; u = 30 [m/s] theta = 45 [deg] quot;This must be commented if you run the parametric table“ quot;Calculationsquot; u_x = u * cos(theta) quot;X-component velocityquot; u_y = u * sin(theta) quot;Y-component velocityquot; t = 2 * u_y / g# quot;Time needed to max distance – from Eq. 1quot; s = u_x * t quot;Max distance – from Eq. 2quot; 38
    • Tutorial 2 :: Projectile Parametric Table o Solve the model by Calculate > Solve menu or pressing F2 and observe the results: 39
    • Tutorial 2 :: Projectile Parametric Table o Now create a Parametric table by adding theta, s, t, ux and uy to it and vary theta from 0 to 90: 40
    • Tutorial 2 :: Projectile Parametric Table o The relation between theta and s can be plotted. 41
    • Plots and Graphs o There are 3 types of graphs that can be plotted with EES:  X-Y plots  Bar plots  X-Y-Z plots (3- dimensional) • Surface plots • Contour plots 42