1. Introducing Error
By: Tala Rifka, Samantha Jones, Anthony Scheller, Zach Palmer, Tyler Droog

2. Introduction and Methodology
Both error and units are necessary, important components of engineering
Unit conversions can cause serious issues, including in some extreme cases, death and bodily harm
Error can also be a factor in calculations that can cause issues
It is important to keep both of these factors in mind when solving engineering problems
Our paper outlines a lot of the concepts of miscoverted units, error and the necessity of both uniformity among units and use of error terms in calculations.

3. Types of Error Conversion Rounding Truncation Human Uncertainty

4. The NASA Disaster
The NASA Mars Climate Orbiter reached Mars on 23rd September to establish an orbit around Mars but it disappeared.
The engineering team used Imperial measurements while the Jet Propulsion Lab Team used the metric system.
As a result, the wrong navigation information was sent to the Orbiter and NASA lost $125 million.

5. Mars Orbiter was doing a aerobraking maneuver to enter into Martian orbit, and when it started it’s rocket motor, it blasted itself out of orbit and crashed.

6. Patriot Missile Failure
On February 25, 1991, during the Gulf War, an American Patriot Missile battery in Dharan, Saudi Arabia, failed to intercept an incoming Iraqi Scud missile. The Scud struck an American Army barracks and killed 28 soldiers
The software truncated the 24-bit numbers
when converting to binary
This truncation error grew the longer the
system ran
At the time of the failure, the system had been
running for over 100 hours

7. Explosion of Ariane 5
On June 4, 1996 an unmanned Ariane 5 rocket launched by the European Space Agency exploded just forty seconds after lift-off
The cause was a software error in the inertial reference system
A 64-bit floating point number was meant to be converted to a 16-bit signed integer
The number was larger than 32,678, which is the largest storable number for 16-bit, so the conversion failed
The development of the rocket cost over $7 billion
The rocket and its cargo cost $500 million

8. Air Canada runs out of fuel
In 1983, an Air Canada plane ran out of fuel in the middle of the flight.
It was Air Canada’s first time to use metric units instead of imperial units but not everybody knew.
Flight 143 after landing at Gimli,Manitoba

9. Medical Mishap
In 1999, a patient was given 0.5 grams of Phenobarbital (a sedative) instead of 0.5 grains (1 grain is about grams).
The Institute for Safe Medication Practices then emphasized that only the metric system can be used for prescribing drugs.

10. Heavy Landing
In 1994, an American International Airways flight landed 15 tons heavier than it should have.
The problem was in kilogram-to-pounds conversion.

11. Columbus’ Arrival
Columbus miscalculated the circumference of the earth as he used Roman miles instead of nautical miles.
This is why he arrived to the Bahamas on October 12, 1492 and thought he arrived to Asia.
unit-conversion-disasters

12. Truncation
Truncation errors occur from using an approximation instead of an exact mathematical procedure.
Example:
derivative of velocity by a finite-difference equation:
dv/dt ~= [v(ti+1)-v(ti)] / (ti+1-ti)

13. Vancouver Stock Exchange
In 1982 the Vancouver Stock Exchange instituted a new index initialized to a value of The index was updated after each transaction. Twenty two months later it had fallen to 520.
The cause was that the updated value was truncated rather than rounded. The rounded calculation gave a value of

14. Round Off
Roundoff errors occur because digital computers cannot represent some quantities exactly. This rounding can lead to erroneous results.
Examples:
Large computations
Floating numbers
Arithmetic operations

15. The sinking of Sleipner A
The Sleipner A platform produces oil and gas in the North Sea at a depth of 82m
The concrete structure that supported the platform sprang a leak and sank
The total economic loss was estimated to be ~$700 million

16. The sinking of Sleipner A ( continued)
The post accident report found the error to an “inaccurate finite element approximation of the linear elastic model of the tricell (using the popular finite element program NASTRAN)” due to rounding errors
The shear stress was underestimated by 47%
The photo on the right shows the crack in the concrete support

17. http://www. google. com/imgres. imgurl=&imgrefurl=http%3A%2F%2Fwww

18. Derived Units
Length, mass and time are the most basic dimensions that allow for the composition of other units.
All other units can be derived from these basic dimensions.

19. Derivations
Derivations arise from need: for example Newton’s law states that F = mass*acceleration.
Mass has units of kilograms (kg).
Acceleration has units of meters per second squared (m/s^2).
The resulting unit of Force is equivalent to a kilogram meter per second squared (kg*m/s^2).
To help shorten the names of units, specific combinations of units can be called something else.
For example, 1 kilogram meter per second squared (kg*m/s^2) is equal to 1 Newton (N).

20. Derived Units
There are 2 different unit systems English and SI (Systeme International).
There are conversions needed to get from one system to the other
There are issues associated with going between systems.
chem105.tripod. com/id6.html

21. Direct and Functional Unit-Conversion Factors
Issues between U.S. and U.K. units
Both use English system
There is a need to be specific when converting to units like gallons (gal)
There are imperial gallons(U.K. units) and regular gallons (gal) that the U.S. uses.
magnusbirkner .blogspot. com

22. Examples of conversions

23. Examples continued
math.tutorvista.com

24. SI Base Units
www. boundless. com
www. processassociates. com

25. Types of conversion factors
Two types of conversion factors:
Direct unit transformation factors
Functional unit transformations (or affine transformations)

26. Direct Unit Transformation Factors
Unit conversions usually require applying an equation, or the multiplication a series of conversion factors.
An example is the conversion between the absolute temperatures Kelvin and Rankin
T(R)= 1.8*T(K)
This is the most common way used in unit conversions

27. Functional Unit Transformations
These require certain definitions such as some temperature unit conversions and the conversion between absolute and gage pressure.
T(0 C) = T(K) T(0F) = T(R)
P(psia)= P(psig)+ 14.7
www. 101qs. com

28. Functional unit transformations (continued)
Some functional unit transformations are more complicated than just a simple addition
An example is the conversion between the gas volume (ft3) and the gas volume at standard state(scf or standard ft3) which depends on the gas conditions and thus requires the use of the real gas equation of state
PV= ZnRT
where P is the absolute pressure, V is the gas volume, Z is the real gas deviation factor, n is the number of moles, R is the universal gas constant and T is the absolute temperature

29. Steps for converting units
Substitute the desired units.
Multiply each variable by unit conversion factors to convert them to the prescribed units of the equation.
Combine the various conversion factors into lumped factors.
Round the answer off to the correct number of significant figures.
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30. Example: Mean hydraulic diameter
The mean hydraulic diameter D (m) of capillary flow paths in porous materials in SI units can be found by
where τ is the tortuosity (dimensionless), k is the permeability in m2,and ϕ is the porosity of the porous media. We want to convert this equation to units of μm2 for the mean hydraulic diameter, permeability in millidarcy and porosity as a percentage.

31. Mean Hydraulic Diameter example continued
After inserting the proper unit-conversion factors in the equation on the previous slide, the following results:
And by combining all the conversion factors:
In order to check the converted equation, we solve it using the desired units and compare the resulting numerical value with the value obtained from the original equation.

32. Common Mistake
A common mistake everyone tends to make is to start with the original units and convert them to the desired units while the opposite is actually correct.
Relating to the example we considered in the previous slide, if this mistake was made we would get the following equation:
Which would simplify to the following result which is incorrect.

33. Matlab Conversion Code
To demonstrate the capabilities of modern technology and applications of Numerical Methods, we have created a simple program in Matlab designed to convert between Celsius, Kelvin, Rankine and Fahrenheit.
The next couple of slides document the program and break it down.

34. Matlab Conversion Code
Figure created by Group 06
Matlab code can be found in the notes below
%Convert temperatures given the original unit, target unit, and numerical
%value of temperature.
%Converts between Fahrenheit(F), Celcius(C), Rankine(R) and Kelvin(K).
%Input is value of temperature, original unit, target unit where the units
%are string values
function tempconvert(t,Orig,Targ)
%Define the temperature string values used to compare the original and
%target temperatures
s1 = 'C';
s2 = 'F';
s3 = 'K';
s4 = 'R';
%Compare the string values of the input
%Will return logical value of 1 or 0 based on whether or not it is true or
%false, respectively
%Compare temperature string with the original temperature
T1 = strcmp(s1,Orig);
T2 = strcmp(s2,Orig);
T3 = strcmp(s3,Orig);
T4 = strcmp(s4,Orig);
%Compare temperature string with the target temperature
T5 = strcmp(s1,Targ);
T6 = strcmp(s2,Targ);
T7 = strcmp(s3,Targ);
T8 = strcmp(s4,Targ);
%Begins conversions
if (T1 == 1 && T5 == 1) || (T2 == 1 && T6 == 1)||(T3 == 1 && T7 == 1)||(T4 == 1 && T8 == 1)
T = t;
elseif T1 == 1 && T5 == 0
if T6 == 1
%Celsius to Fahrenheit
T = 9/5*t+32;
elseif T7 == 1
%Celsius to Kelvin
T = t ;
elseif T8 == 1
%Celsius to Rankine
T = 9/5*(t );
end
elseif T2 == 1 && T6 == 0
if T5 == 1
%Fahrenheit to Celsius
T = (t-32)*5/9;
%Fahrenheit to Kelvin
T = (t-32)*5/ ;
%Fahrenheit to Rankine
T =(t );
elseif T3 == 1 && T7 == 0
%Kelvin to Celsius
T = t ;
elseif T6 == 1
%Kelvin to Fahrenheit
T = t*9/ ;
%Kelvin to Rankine
T = 9/5*t;
elseif T4 == 1 && T8 == 0
%Rankine to Celsius
T = (t )*5/9;
%Rankine to Fahrenheit
T = t ;
%Rankine to Kelvin
T = (5/9)*t;
%Ouput of the converted T (Temperature) and Unit T
Unit = Targ

35. Matlab Conversion Code
Figure created by Group 06

36. Matlab Conversion Code
Figure created by Group 06

37. Outputs & Testing
Figures created by Group 06

38. Output & Testing
Figures created by Group 06

39. Final Remarks
The NASA incident as well as the other examples above that occurred as a result of units could and should have been prevented.
Engineers must take unit conversions seriously and be very careful when handling units.
Clear unit conversions shown and excellent communication are some of the best things that can be done to prevent unit errors.
The most important thing an engineer can do to prevent unit errors is to understand the importance of units and the seriousness of the errors that can occur as a result.
Both units and error are important factors to think about when doing calculations that have the possibility to hurt other people.

40. References/Resources