2.2 Introductory Mathematical Concepts
2.2.1 Units and Unit Systems
2.2.1.3 Consistent Sets of Units and Unit Conversions
Dimensional consistency should not be confused with a consistent set of units, which refers to a unit set that does not require any conversion factors in calculations or in mathematical expressions of fundamental physics. The base units of the SI and English systems are each a consistent set of units. For instance, if we look at the consistent English units for the weight, W, of a mass, m, in a gravitational field with an acceleration of gravity, g, we have
[W] = [m][g] = slug × ft
s2 = lbf (2.1)
where the resultant weight is in the consistent units of pounds force, lbf. If we use inconsistent units of pound mass, lbm, we get
[W] = [m][g] = lbm× ft
s2 = lbmft
s2 (2.2)
To obtain the consistent units of pounds force, we need to convert the pounds mass to slugs, where 32.2 lbmis equal to 1 slug. We would then need to write the weight equation as
W = 1
gcmg (2.3)
where gcis the conversion factor, equal to
gc= 32.2 lbm
slug (2.4)
Equation (2.4) is telling us that a mass of one slug is 32.2 times larger than a pound mass. Using inconsistent units, Equation (2.3) is then given by
[W] = 1
gc[m][g] = 1 gc
( lbm× ft
s2 )
= lbf (2.5)
In SI units, the conversion factor is
gc= 9.81 kgf
N (2.6)
k
k k
Introductory Concepts 103
If we use this conversion factor for our weight in SI units, we obtain the inconsistent SI unit for weight of kilogram force, kgf.
[W] = 1
gc[m][g] = 1 gc
( kg ×m
s2 )
= kgf (2.7)
From a certain perspective, the use of inconsistent units in Equations (2.5) and (2.7) makes some sense. In each case, the unit weight is the same as the unit of mass. In Equation (2.5), a pound of weight equals a pound of mass, and in Equation (2.7) a kilogram of weight equals a kilogram of mass. However, from an engineering perspective this makes things more complicated and prone to errors, usually by a factor of 32.2 or 9.81. The bottom line is that we should always strive to use consistent units and avoid the addition of these inconsistent unit conversion factors in our equations.
That said, be aware that inconsistent units, such as the pound mass, lbm, are still found frequently in engineering, especially in the fields thermodynamics and propulsion.
As a clarification of unit symbols used in this text, we simply use lb, rather than lbf, to denote a pound force, lbmto denote a pound mass, and kg to denote a kilogram mass. We will not find much occasion to use the unit of kilogram force, kgf.
In regards to temperature, the kelvin, in the SI system, and the degree Rankine, in the English sys-tem, represent consistent units based on absolute temperature scales. In an absolute temperature scale, the “bottom” of the scale corresponds to absolute zero, where, theoretically, all molecu-lar translational motion ceases. Therefore, zero kelvin, 0 K, and zero degrees Rankine, 0∘R, are equivalent, corresponding to absolute zero.
We often deal with temperatures in units of degrees Fahrenheit and degrees Celsius, which are not based on absolute temperature scales. Conversions from units of degrees Celsius and degrees Fahrenheit to consistent units are given below.
K = ∘C + 273.15 (2.8)
∘R = ∘F + 459.67 (2.9)
Based on these equations, we see that 0 K = −273.15∘C and 0∘R = −459.67∘F. Conversely, 0∘C = 273.15K and 0∘F = 459.67∘R. Conversions between Celsius and Fahrenheit are as follows.
∘C = 5
9(∘F − 32) (2.10)
∘F = 9
5∘C + 32 (2.11)
Again, be prepared to properly convert temperature to consistent units, as the inconsistent units of Centigrade and Fahrenheit are still widely used.
Finally, as a natural consequence of dealing with all of these different sets of units, we must be able to perform unit conversions within the same unit system and between different unit systems.
In performing calculations, we usually want to convert to the set of units that are appropriate for the quantities being calculated. For example, if we are calculating the time it takes to fly from Los Angeles to London, it would be more appropriate for us to use units of hours instead of seconds. If we are recording the airspeed of an aircraft, we are probably writing down knots or miles per hour from an airspeed indicator, rather than feet per second. We often find it necessary to perform many unit conversions in a calculation. (A list of useful unit conversion factors is given in Appendix B.) Sometimes, it is beneficial to convert all or most of the given quantities into consistent units prior to starting the calculation. Nevertheless, it is best to carry your units through the entire calculation to minimize errors and to help check the validity of the results.
The following two examples illustrate the importance of units.
k k Example 2.1 The Importance of Units: The “Gimli” Glider On 23 July 1983, Air Canada
Flight 143, a twin-engine Boeing 767 commercial airliner, departed Ottawa, Canada with a planned destination of Edmonton, Canada. About an hour into the flight, at a cruising altitude of 41,000 ft (12,500 m), both turbofan engines “flamed out”. The Boeing 767 airliner had inexplicably run out of fuel. Luckily, there was a decommissioned Canadian air force base within gliding distance of the aircraft, in Gimli, Manitoba, Canada. The airliner successfully landed at the Gimli airfield, collapsing the nose landing gear, which could only be partially extended due to the power loss of both engines. After the landing, the powerless airliner was dubbed the “Gimli Glider”. So, how did this advanced jet airliner simply run out of fuel? As with most aviation incidents and accidents, there was a chain of events that led up to this potentially catastrophic event.
The aircraft needed to be fueled for the flight from Ottawa to Edmonton, but unlike “filling up”
your automobile, an airliner is not necessarily just “filled up”. If the airliner carries more fuel than is required for a flight, it is carrying excess weight, which results in a performance and cost penalty. So, the required fuel quantity, which includes an extra amount of fuel called a reserve, is calculated for each flight. A total of 22,300 kg of fuel was required for the Ottawa to Edmonton flight. The night before the flight, the Boeing 767’s computerized fuel indication and monitoring system had failed. Rather than using this computerized system, the amount of fuel, to be added, was calculated manually. It was determined that the aircraft had 7682 liters of fuel in its tanks prior to adding any fuel. This fuel quantity was converted from liters to kilograms, using a conversion factor of 1.77 kg/liter, as follows.
7682 liters × 1.77 kg
liter = 13,597kg
Subtracting this amount of fuel in the aircraft fuel tanks from the amount of fuel required for the flight, the fuel quantity to be added was calculated as
22,300kg − 13,597kg = 8703kg
Since the fuel was added to the aircraft from a fuel truck, which dispensed fuel in liters rather than kilograms, the fuel quantity to be added was converted from kilograms to liters, as
8703 kg
1.77literkg = 4907 liters
The problem with the above calculations is that an incorrect value for the liters-kilogram conver-sion was used. The correct converconver-sion is 0.8 kg/liter rather than 1.77 kg/liter. Using the correct conversion, the actual fuel quantity in the aircraft tanks prior to adding fuel was
7682 liters × 0.8 kg
liter = 6146 kg The amount of fuel that needed to be added should have been
22,300kg − 6146kg = 16,154kg Thus, the amount of fuel in liters that should have been added was
8703 kg
0.8literkg = 10,879 liters
Using the incorrect conversion, the aircraft took off with a total of only 12,589 liters of fuel in its fuel tanks, rather than the required 22,300 liters.
k
k k
Introductory Concepts 105
The reason for the use of the incorrect conversion value was not just a matter of an erroneous number, but was also due to an error in units. The Boeing 767 was a new addition to the Air Canada fleet, bringing with it several advancements in computerized control of the aircraft sys-tems. However, these advancements came with some changes in the normal procedures that had been followed with the other aircraft in the Air Canada fleet. The Boeing 767 was the first air-craft in their fleet which measured fuel in SI units of kilograms, rather than English units of pounds. Prior to the introduction of the Boeing 767, the fuel quantities for Air Canada aircraft were converted from pounds to liters, using the correct conversion factor of 1.77 lb/liter. Hence, this same value was erroneously used for the calculation of fuel, in kilograms, for the new Boeing 767.
Example 2.2 The Importance of Units: The Mars Climate Orbiter The Mars Climate Orbiter (MCO), shown in Figure 2.2, and the Mars Polar Lander (MPL) were part of a series of NASA spacecraft missions to explore Mars in the late 1990s. The MCO spacecraft bus (the bus is the spacecraft platform or modular infrastructure upon which the payload or experiments and instru-mentation are mounted) dimensions were approximately 2.1 m (6.9 ft) tall, 1.6 m (5.2 ft) wide, and 2.0 m (6.6 ft) deep with a launch weight of 338 kg (745 lb). The fully extended solar panel array measured 5.5 m (18 ft) in length. The MCO and MPL total mission cost was $327.6 million which included $193.1 million for spacecraft development, $91.7 million for launch services, and $42.8 million for operations.
The MCO spacecraft was launched from Cape Canaveral, Florida, aboard a Delta II launch vehicle, on 11 December 1998. The MPL was launched from Cape Canaveral on 3 January 1999
High gain antenna
Star trackers
Battery enclosureUHF Antenna
Mars color imaging (MARCI) system
Leros main engine (for mars orbit insertion only)
Pressure modulated infrared radiometer
(PMIRR) Hydrazine thruster Solar array
Figure 2.2 The Mars Climate Orbiter spacecraft. (Source: NASA.)
k k aboard a similar Delta II rocket. After a nine and a half month, 416 million mile (669 million km)
journey to Mars, the MCO was to enter Mars orbit and remain there to collect long-term atmo-spheric and weather data and to serve as a communications relay for the MPL, which would land on the surface of Mars.
On arrival to Mars, the MCO was to perform an orbital insertion burn (fire its main engine to decelerate) and enter an elliptical orbit about the planet. The spacecraft would then use aerobrak-ing, “dipping” in and out of the Martian atmosphere, creating atmospheric drag that would slow the vehicle and circularize its orbit. On 23 September 1999, shortly after entering the Martian atmosphere on a much lower than planned trajectory, all communication with the MCO was lost.
Communication with the MCO was never reestablished and the spacecraft was presumed lost.
An MCO Mishap Investigation Board (MIB) was established to investigate the loss of the space-craft. It was discovered that the spacecraft had entered Mars orbit with a periapsis (the lowest altitude of its orbit) of 57 km, when this lowest altitude should have been a much higher 226 km. At the lower entry altitude, the spacecraft encountered a much denser region of the Martian atmo-sphere, resulting in either intolerable atmospheric drag and the subsequent destruction of the spacecraft, as it descended further into the atmosphere, or “skipping” of the vehicle out of the atmosphere, sending it into an orbit around the Sun. The lowest survivable altitude was deter-mined to be about 80 km. So, why was the periapsis 170 km lower than expected, resulting in the loss of the MCO?
To fully understand the cause of the mishap, we need to know a little more about the MCO spacecraft attitude and trajectory control. During its nine-month journey to Mars, the spacecraft’s attitude and trajectory was controlled using eight small hydrazine monopropellant thrusters and three reaction wheels. As is typical of reaction wheel systems, excess momentum built up in the MCO wheel system due to external torques, such as from Sun-induced pressure on the solar panel array. To remove this excess angular momentum, the MCO thrusters were fired periodically during its nine-month spaceflight, in what are called angular momentum desaturation (AMD) maneuvers.
The required attitude and trajectory corrections were calculated using ground-based computer software over the course of the nine-month spaceflight. Two relevant pieces of software used for the corrections, to be discussed in the root cause of the mishap, included the software that calcu-lated the thruster forces (software file “Small Forces”) and the software that used these thruster force numbers to calculate the spacecraft attitude corrections (software file “Angular Momentum Desaturation”).
Given this background information, we now review the single root cause of the MCO mishap, as determined by the MCO Mishap Investigation Board, cited below.
The MCO MIB has determined that the root cause for the loss of the MCO space-craft was the failure to use metric units in the coding of a ground software file,
“Small Forces,” used in trajectory models. Specifically, thruster performance data in English units instead of metric units was used in the software application code titled SM_FORCES (small forces). The output from the SM_FORCES application code as required by a MSOP (Mars Surveyor Operations Project) Software Interface Specification (SIS) was to be in metric units of newton-seconds (N-s). Instead, the data was reported in English units of pound-seconds (lbf-s). The Angular Momentum Desaturation (AMD) file contained the output data from the SM_FORCES software.
The SIS, which was not followed, defines both the format and units of the AMD file generated by ground-based computers. Subsequent processing of the data from the AMD file by the navigation software algorithm therefore, underestimated the effect on the spacecraft trajectory by a factor of 4.45, which is the required conversion factor
k
k k
Introductory Concepts 107
from force in pounds to newtons. An erroneous trajectory was computed using this incorrect data.
Excerpt from MCO MIB Phase I Report, 10 November 1999 ([31]) This unit conversion error resulted in several small errors that accumulated, over the nine-month trip of the Mars Climate Orbiter, into a larger, ultimately catastrophic, error in the trajectory. The bottom line is that a “simple” unit conversion error, not converting the thruster force from English units to metric units, led to the demise of a multi-million dollar space probe. Needless to say, careful attention to units is important!