1. Engineering Equation Solver
Denpong Soodphakdee, Ph.D.
Department of Mechanical Engineering
Khon Kaen University
denpong@kku.ac.th
2. EES ME KKU License
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3. 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
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4. 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!
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5. 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.
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6. 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
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7. 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
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8. 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)
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9. 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!
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10. 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!
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12. 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)
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13. 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
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14. The Unit System
o The unit system can be set by
Option > Unit System
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15. 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)
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16. The Unit System
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17. 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]
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18. 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
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19. 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)
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20. 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)
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21. 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;
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22. 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)
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23. 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)
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24. Built-in Functions
o Solid property example
quot;Young’s Modulus – note the underscorequot;
T = 140 [C]
E = E_(Aluminum, T)
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25. Built-in Functions
o The property
functions can be
pasted from menu
Option > Function
Information
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26. 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.
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27. END OF EES LECTURE 1
28. 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.
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29. 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
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30. 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
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31. 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:
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32. The Parametric Table
o The simple equation can be solved for
each value of theta and the results y
are displayed in the table.
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33. 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
•
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34. Plot Basics
o Create x-y plot from Plots menu:
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35. Plot Basics
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36. 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.
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37. 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;
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38. 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;
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39. Tutorial 2 :: Projectile Parametric Table
o Solve the model by Calculate > Solve
menu or pressing F2 and observe the
results:
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40. 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:
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41. Tutorial 2 :: Projectile Parametric Table
o The relation between theta and s can
be plotted.
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42. 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
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